Chapter 8 Roots and Radicals


 Arabella Waters
 1 years ago
 Views:
Transcription
1 Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi Commons) Chpter Outline 8. Simplify Expressions with Roots 8. Simplify Rdil Expressions 8. Simplify Rtionl Exponents 8. Add, Sutrt, nd Multiply Rdil Expressions 8. Divide Rdil Expressions 8.6 Solve Rdil Equtions 8.7 Use Rdils in Funtions 8.8 Use the Complex Numer System Introdution Imgine hrging your ell phone is less thn five seonds. Consider lening rdiotive wste from ontminted wter. Think out filtering slt from oen wter to mke n endless supply of drinking wter. Ponder the ide of ioni devies tht n repir spinl injuries. These re just of few of the mny possile uses of mteril lled grphene. Mterils sientists re developing mteril mde up of single lyer of ron toms tht is stronger thn ny other mteril, ompletely flexile, nd onduts eletriity etter thn most metls. Reserh into this type of mteril requires solid kground in mthemtis, inluding understnding roots nd rdils. In this hpter, you will lern to simplify expressions ontining roots nd rdils, perform opertions on rdil expressions nd equtions, nd evlute rdil funtions. 8. Simplify Expressions with Roots Lerning Ojetives By the end of this setion, you will e le to: Simplify expressions with roots Estimte nd pproximte roots Simplify vrile expressions with roots Be Prepred! Before you get strted, tke this rediness quiz.. Simplify: ( 9) 9 ( 9). If you missed this prolem, review Exmple..
2 7 Chpter 8 Roots nd Rdils. Round.86 to the nerest hundredth. If you missed this prolem, review Exmple... Simplify: x x y y y z z z z. If you missed this prolem, review Exmple.. Simplify Expressions with Roots In Foundtions, we riefly looked t squre roots. Rememer tht when rel numer n is multiplied y itself, we write n nd red it n squred. This numer is lled the squre of n, nd n is lled the squre root. For exmple, Squre nd Squre Root of numer is red squred 69 is lled the squre of, sine = 69 is squre root of 69 Squre Squre Root If n = m, then m is the squre of n. If n = m, then n is squre root of m. Notie ( ) = 69 lso, so is lso squre root of 69. Therefore, oth nd re squre roots of 69. So, every positive numer hs two squre roots one positive nd one negtive. Wht if we only wnted the positive squre root of positive numer? We use rdil sign, nd write, m, whih denotes the positive squre root of m. The positive squre root is lso lled the prinipl squre root. We lso use the rdil sign for the squre root of zero. Beuse 0 = 0, root. Squre Root Nottion 0 = 0. Notie tht zero hs only one squre m is red the squre root of m. If n = m, then n = m, for n 0. We know tht every positive numer hs two squre roots nd the rdil sign indites the positive one. We write 69 =. If we wnt to find the negtive squre root of numer, we ple negtive in front of the rdil sign. For exmple, 69 =. EXAMPLE 8. Simplify: 89. Sine =. Sine 7 = 89 nd the negtive is in front of the rdil sign This OpenStx ook is ville for free t
3 Chpter 8 Roots nd Rdils 7 TRY IT : : 8. Simplify: 6. TRY IT : : 8. Simplify: 00. Cn we simplify 9? Is there numer whose squre is 9? ( ) = 9 Any positive numer squred is positive. Any negtive numer squred is positive. There is no rel numer equl to 9. The squre root of negtive numer is not rel numer. EXAMPLE 8. Simplify: There is no rel numer whose squre is 96. The negtive is in front of the rdil is not rel numer. 6 8 TRY IT : : 8. Simplify: TRY IT : : 8. Simplify: 9. So fr we hve only tlked out squres nd squre roots. Let s now extend our work to inlude higher powers nd higher roots. Let s review some voulry first. We write: n n n n We sy: n squred n ued n to the fourth power n to the fi th power The terms squred nd ued ome from the formuls for re of squre nd volume of ue. It will e helpful to hve tle of the powers of the integers from to. See Figure 8.. Figure 8. Notie the signs in the tle. All powers of positive numers re positive, of ourse. But when we hve negtive numer, the even powers re positive nd the odd powers re negtive. We ll opy the row with the powers of to help you see
4 76 Chpter 8 Roots nd Rdils this. We will now extend the squre root definition to higher roots. n th Root of Numer If n =, then is n n th root of. The prinipl n th root of is written n. n is lled the index of the rdil. Just like we use the word ued for, we use the term ue root for We n refer to Figure 8. to help find higher roots. = 6 = 8 ( ) =. 6 = 8 = = Could we hve n even root of negtive numer? We know tht the squre root of negtive numer is not rel numer. The sme is true for ny even root. Even roots of negtive numers re not rel numers. Odd roots of negtive numers re rel numers. Properties of n When n is n even numer nd 0, then n is rel numer. < 0, then n When n is n odd numer, is not rel numer. n is rel numer for ll vlues of. We will pply these properties in the next two exmples. EXAMPLE 8. Simplify: Sine = 6. 8 Sine () = 8. Sine () =. This OpenStx ook is ville for free t
5 Chpter 8 Roots nd Rdils 77 TRY IT : : 8. Simplify: 7 6. TRY IT : : 8.6 Simplify: In this exmple e lert for the negtive signs s well s even nd odd powers. EXAMPLE 8. Simplify: Sine ( ) =. 6. Think, (?) = 6. No rel numer rised to the fourth power is negtive. Sine ( ) =. 6 Not rel numer. TRY IT : : 8.7 Simplify: 7 6. TRY IT : : 8.8 Simplify: Estimte nd Approximte Roots When we see numer with rdil sign, we often don t think out its numeril vlue. While we proly know tht the =, wht is the vlue of or 0? In some situtions quik estimte is meningful nd in others it is onvenient to hve deiml pproximtion. To get numeril estimte of squre root, we look for perfet squre numers losest to the rdind. To find n estimte of, we see is etween perfet squre numers 9 nd 6, loser to 9. Its squre root then will e etween nd, ut loser to. Similrly, to estimte 9, we see 9 is etween perfet ue numers 6 nd. The ue root then will e etween nd.
6 78 Chpter 8 Roots nd Rdils EXAMPLE 8. Estimte eh root etween two onseutive whole numers: 0. Think of the perfet squre numers losest to 0. Mke smll tle of these perfet squres nd their squres roots. Lote 0 etween two onseutive perfet squres. 0 is etween their squre roots. Similrly we lote etween two perfet ue numers. Lote etween two onseutive perfet ues. is etween their ue roots. TRY IT : : 8.9 Estimte eh root etween two onseutive whole numers: 8 9 TRY IT : : 8.0 Estimte eh root etween two onseutive whole numers: 8 There re mthemtil methods to pproximte squre roots, ut nowdys most people use lultor to find squre roots. To find squre root you will use the x key on your lultor. To find ue root, or ny root with higher index, y you will use the x key. When you use these keys, you get n pproximte vlue. It is n pproximtion, urte to the numer of digits shown on your lultor s disply. The symol for n pproximtion is nd it is red pproximtely. Suppose your lultor hs 0 digit disply. You would see tht rounded to two deiml ples is rounded to two deiml ples is 9. This OpenStx ook is ville for free t
7 Chpter 8 Roots nd Rdils 79 How do we know these vlues re pproximtions nd not the ext vlues? Look t wht hppens when we squre them: ( ) = (.) =.076 (.0799) = (.) = 9.98 Their squres re lose to, ut re not extly equl to. The fourth powers re lose to 9, ut not equl to 9. EXAMPLE 8.6 Round to two deiml ples: Use the lultor squre root key..066 Round to two deiml ples.. y Use the lultor x key Round to two deiml ples..66 y Use the lultor x key Round to two deiml ples TRY IT : : 8. Round to two deiml ples: 7 7. TRY IT : : 8. Round to two deiml ples: Simplify Vrile Expressions with Roots The odd root of numer n e either positive or negtive. For exmple, But wht out n even root? We wnt the prinipl root, so 6 But notie, =.
8 70 Chpter 8 Roots nd Rdils How n we mke sure the fourth root of rised to the fourth power is? We n use the solute vlue. =. So n we sy tht when n is even n =. This gurntees the prinipl root is positive. Simplifying Odd nd Even Roots For ny integer n, when the index n is odd when the index n is even We must use the solute vlue signs when we tke n even root of n expression with vrile in the rdil. EXAMPLE 8.7 n n n n = = Simplify: x n p d y. We use the solute vlue to e sure to get the positive root. n Sine the index n is even, n x =. x This is n odd indexed root so there is no need for n solute vlue sign. n Sine the index n is odd, n n Sine the index n is even n d n Sine the index n is odd, n m =. m p =. p y =. y TRY IT : : 8. Simplify: w m d q. TRY IT : : 8. Simplify: y p z d q. Wht out squre roots of higher powers of vriles? The Power Property of Exponents sys ( m ) n = m n. So if we squre m, the exponent will eome m. ( m ) = m This OpenStx ook is ville for free t
9 Chpter 8 Roots nd Rdils 7 Looking now t the squre root, m Sine ( m ) = m. ( m ) n Sine n is even n =. m We pply this onept in the next exmple. EXAMPLE 8.8 Simplify: x 6 y 6. Sine x = x 6. So m = m. x 6 x Sine the index n is even n =. x Sine y 8 = y 6. y 6 y 8 n Sine the index n is even n In this se the solute vlue sign is not needed s y 8 is positive. =. y 8 TRY IT : : 8. Simplify: y 8 z. TRY IT : : 8.6 Simplify: m 0. The next exmple uses the sme ide for highter roots. EXAMPLE 8.9 Simplify: y 8 z 8. Sine y 6 = y 8. y 8 y 6 n Sine n is odd, n =. y 6
10 7 Chpter 8 Roots nd Rdils Sine z = z 8. z 8 z Sine z is positive, we do not need n z solute vlue sign. TRY IT : : 8.7 Simplify: u v. TRY IT : : 8.8 Simplify: 0 6 d In the next exmple, we now hve oeffiient in front of the vrile. The onept wy. m = m works in muh the sme 6r = r euse r = 6r. But notie u 8 = u nd no solute vlue sign is needed s u is lwys positive. EXAMPLE 8.0 Simplify: 6n 8. 6n Sine (n) = 6n. (n) n Sine the index n is even n =. n 8 Sine (9) = 8. (9) n Sine the index n is even n =. 9 TRY IT : : 8.9 Simplify: 6x 00p. TRY IT : : 8.0 Simplify: 69y y. This exmple just tkes the ide frther s it hs roots of higher index. EXAMPLE 8. Simplify: 6p 6 6q. This OpenStx ook is ville for free t
11 Chpter 8 Roots nd Rdils 7 6p 6 Rewrite 6p 6 s p. p Tke the ue root. p Rewrite the rdind s fourth power. 6q q Tke the fourth root. q TRY IT : : 8. Simplify: 7x 7 8q 8. TRY IT : : 8. Simplify: q 9 q. The next exmples hve two vriles. EXAMPLE 8. Simplify: 6x y 6 8 6p 6 q 9. 6x y Sine 6xy = 6x y 6xy Tke the squre root. 6 xy 6 8 Sine = 6 8 Tke the squre root. 6p 6 q 9 Sine p q = 6p 6 q 9 p q Tke the ue root. p q
12 7 Chpter 8 Roots nd Rdils TRY IT : : 8. Simplify: 00 p q 0 8x 0 y TRY IT : : 8. Simplify: m n 69x 0 y 7w 6 z MEDIA : : Aess this online resoure for dditionl instrution nd prtie with simplifying expressions with roots. Simplifying Vriles Exponents with Roots using Asolute Vlues ( 7SimVrAVl) This OpenStx ook is ville for free t
13 Chpter 8 Roots nd Rdils 7 8. EXERCISES Prtie Mkes Perfet Simplify Expressions with Roots In the following exerises, simplify Estimte nd Approximte Roots In the following exerises, estimte eh root etween two onseutive whole numers In the following exerises, pproximte eh root nd round to two deiml ples Simplify Vrile Expressions with Roots In the following exerises, simplify using solute vlues s neessry. 7. u 8 v y 7 m k 8 6 p 6. x 6 y 6. w. x y. 6. x 9 y m 8 n r s 0
14 76 Chpter 8 Roots nd Rdils 9. 9x 8x y 00m. m x 6 x. 6x 8 6 6y. 8 9 d p q 6 7p q r 8s d 6x y x y 69w 8 y x y z 6r 6 s 0 y 8 z 7 Writing Exerises. Why is there no rel numer equl to 6?. Wht is the differene etween 9 nd 9?. Explin wht is ment y the n th root of numer.. Explin the differene of finding the n th root of numer when the index is even ompred to when the index is odd. Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. If most of your heks were: onfidently. Congrtultions! You hve hieved the ojetives in this setion. Reflet on the study skills you used so tht you n ontinue to use them. Wht did you do to eome onfident of your ility to do these things? Be speifi. with some help. This must e ddressed quikly euse topis you do not mster eome potholes in your rod to suess. In mth every topi uilds upon previous work. It is importnt to mke sure you hve strong foundtion efore you move on. Who n you sk for help? Your fellow lssmtes nd instrutor re good resoures. Is there ple on mpus where mth tutors re ville? Cn your study skills e improved? no  I don t get it! This is wrning sign nd you must not ignore it. You should get help right wy or you will quikly e overwhelmed. See your instrutor s soon s you n to disuss your sitution. Together you n ome up with pln to get you the help you need. This OpenStx ook is ville for free t
15 Chpter 8 Roots nd Rdils Simplify Rdil Expressions Lerning Ojetives By the end of this setion, you will e le to: Use the Produt Property to simplify rdil expressions Use the Quotient Property to simplify rdil expressions Be Prepred! Before you get strted, tke this rediness quiz.. Simplify: x 9 x. If you missed this prolem, review Exmple... Simplify: y y. If you missed this prolem, review Exmple... Simplify: n 6. If you missed this prolem, review Exmple.7. Use the Produt Property to Simplify Rdil Expressions We will simplify rdil expressions in wy similr to how we simplified frtions. A frtion is simplified if there re no ommon ftors in the numertor nd denomintor. To simplify frtion, we look for ny ommon ftors in the numertor nd denomintor. A rdil expression, n, is onsidered simplified if it hs no ftors of m n. So, to simplify rdil expression, we look for ny ftors in the rdind tht re powers of the index. Simplified Rdil Expression For rel numers nd m, nd n, n is onsidered simplified i hs no ftors of m n For exmple, is onsidered simplified euse there re no perfet squre ftors in. But is not simplified euse hs perfet squre ftor of. Similrly, perfet ue ftor of 8. is simplified euse there re no perfet ue ftors in. But is not simplified euse hs To simplify rdil expressions, we will lso use some properties of roots. The properties we will use to simplify rdil expressions re similr to the properties of exponents. We know tht () n = n n. The orresponding of Produt n Property of Roots sys tht = n n. Produt Property of n th Roots If n n nd re rel numers, nd n is n integer, then n = n n nd n n n = We use the Produt Property of Roots to remove ll perfet squre ftors from squre root. EXAMPLE 8. SIMPLIFY SQUARE ROOTS USING THE PRODUCT PROPERTY OF ROOTS Simplify: 98.
16 78 Chpter 8 Roots nd Rdils TRY IT : : 8. Simplify: 8. TRY IT : : 8.6 Simplify:. Notie in the previous exmple tht the simplified form of 98 is 7, whih is the produt of n integer nd squre root. We lwys write the integer in front of the squre root. 7 Be reful to write your integer so tht it is not onfused with the index. The expression 7 is very different from. HOW TO : : SIMPLIFY A RADICAL EXPRESSION USING THE PRODUCT PROPERTY. Step. Step. Step. Find the lrgest ftor in the rdind tht is perfet power of the index. Rewrite the rdind s produt of two ftors, using tht ftor. Use the produt rule to rewrite the rdil s the produt of two rdils. Simplify the root of the perfet power. We will pply this method in the next exmple. It my e helpful to hve tle of perfet squres, ues, nd fourth powers. EXAMPLE 8. Simplify: Rewrite the rdind s produt using the lrgest perfet squre ftor. 00 Rewrite the rdil s the produt of two rdils 00 Simplify. 0 This OpenStx ook is ville for free t
17 Chpter 8 Roots nd Rdils 79 Rewrite the rdind s produt using the gretest perfet ue ftor. = 8 Rewrite the rdil s the produt of two rdils Simplify. Rewrite the rdind s produt using the gretest perfet fourth power ftor. = 8 Rewrite the rdil s the produt of two rdils 8 8 Simplify. TRY IT : : 8.7 Simplify: TRY IT : : 8.8 Simplify: The next exmple is muh like the previous exmples, ut with vriles. Don t forget to use the solute vlue signs when tking n even root of n expression with vrile in the rdil. EXAMPLE 8. Simplify: x x x 7. Rewrite the rdind s produt using the lrgest perfet squre ftor. Rewrite the rdil s the produt of two rdils. Simplify. Rewrite the rdind s produt using the lrgest perfet ue ftor. Rewrite the rdil s the produt of two rdils. Simplify. x x x x x x x x x x. x x x x
18 760 Chpter 8 Roots nd Rdils x 7 Rewrite the rdind s produt x x using the gretest perfet fourth power ftor. Rewrite the rdil s the produt of two x x rdils. Simplify. x x TRY IT : : 8.9 Simplify: y 6 z TRY IT : : 8.0 Simplify: p 9 y 8 6 q We follow the sme proedure when there is oeffiient in the rdind. In the next exmple, oth the onstnt nd the vrile hve perfet squre ftors. EXAMPLE 8.6 Simplify: 7n 7 x 7 80y. Rewrite the rdind s produt using the lrgest perfet squre ftor. Rewrite the rdil s the produt of two rdils. Simplify. 7n 7 6n 6 n 6n 6 n 6 n n Rewrite the rdind s produt using perfet ue ftors. Rewrite the rdil s the produt of two rdils. x 7 8x 6 x 8x 6 x Rewrite the fir t rdind s x. x Simplify. x x x This OpenStx ook is ville for free t
19 Chpter 8 Roots nd Rdils 76 Rewrite the rdind s produt using perfet fourth power ftors. Rewrite the rdil s the produt of two rdils. Rewrite the fir t rdind s y. y 80y 6y y 6y y y Simplify. y y TRY IT : : 8. Simplify: y p 0 6q 0. TRY IT : : 8. Simplify: 7 9 8m 6n 7. In the next exmple, we ontinue to use the sme methods even though there re more thn one vrile under the rdil. EXAMPLE 8.7 Simplify: 6u v 0x y 8x y 7. Rewrite the rdind s produt using the lrgest perfet squre ftor. Rewrite the rdil s the produt of two rdils. 6u v 9u v 7uv 9u v 7uv Rewrite the fir t rdind s uv. uv 7uv Simplify. u v 7uv Rewrite the rdind s produt using the lrgest perfet ue ftor. Rewrite the rdil s the produt of two rdils. 0x y 8x y xy 8x y xy Rewrite the fir t rdind s xy. xy xy Simplify. xy xy
20 76 Chpter 8 Roots nd Rdils 8x y 7 Rewrite the rdind s produt using the lrgest perfet fourth power ftor. Rewrite the rdil s the produt of two rdils. Rewrite the fir t rdind s xy. xy 6x y y 6x y y y Simplify. xy y TRY IT : : 8. Simplify: x y x y 8. TRY IT : : 8. Simplify: 80m 9 n 7x 6 y 80x 7 y. EXAMPLE 8.8 Simplify: 7 6. Rewrite the rdind s produt using perfet ue ftors. Tke the ue root. There is no rel numer n where n = 6. 7 ( ) 6 Not rel numer. TRY IT : : 8. Simplify: 6 8. TRY IT : : 8.6 Simplify: 6. We hve seen how to use the order of opertions to simplify some expressions with rdils. In the next exmple, we hve the sum of n integer nd squre root. We simplify the squre root ut nnot dd the resulting expression to the integer sine one term ontins rdil nd the other does not. The next exmple lso inludes frtion with rdil in the numertor. Rememer tht in order to simplify frtion you need ommon ftor in the numertor nd denomintor. EXAMPLE 8.9 Simplify: + 8. This OpenStx ook is ville for free t
21 Chpter 8 Roots nd Rdils 76 + Rewrite the rdind s produt using the lrgest perfet squre ftor. + 6 Rewrite the rdil s the produt of two rdils. + 6 Simplify. + The terms nnot e dded s one hs rdil nd the other does not. Trying to dd n integer nd rdil is like trying to dd n integer nd vrile. They re not like terms! Rewrite the rdind s produt using the lrgest perfet squre ftor. Rewrite the rdil s the produt of two rdils Simplify. Ftor the ommon ftor from the numertor. Remove the ommon ftor,, from the numertor nd denomintor. Simplify. TRY IT : : 8.7 Simplify: TRY IT : : 8.8 Simplify: Use the Quotient Property to Simplify Rdil Expressions Whenever you hve to simplify rdil expression, the first step you should tke is to determine whether the rdind is perfet power of the index. If not, hek the numertor nd denomintor for ny ommon ftors, nd remove them. You my find frtion in whih oth the numertor nd the denomintor re perfet powers of the index. EXAMPLE 8.0 Simplify: Simplify inside the rdil fir t. Rewrite showing the ommon ftors of the numertor nd denomintor. Simplify the frtion y removing ommon ftors. Simplify. Note =
22 76 Chpter 8 Roots nd Rdils Simplify inside the rdil fir t. Rewrite showing the ommon ftors of the numertor nd denomintor. Simplify the frtion y removing ommon ftors. Simplify. Note = 8 7. Simplify inside the rdil fir t. Rewrite showing the ommon ftors of the numertor nd denomintor. Simplify the frtion y removing ommon ftors. Simplify. Note = 6. TRY IT : : 8.9 Simplify: TRY IT : : 8.0 Simplify: In the lst exmple, our first step ws to simplify the frtion under the rdil y removing ommon ftors. In the next exmple we will use the Quotient Property to simplify under the rdil. We divide the like ses y sutrting their exponents, m n = m n, 0 EXAMPLE 8. Simplify: m6 m 8 0. Simplify the frtion inside the rdil fir t. Divide the like ses y sutrting the exponents. Simplify. m 6 m m m This OpenStx ook is ville for free t
23 Chpter 8 Roots nd Rdils 76 8 Use the Quotient Property of exponents to simplify the frtion under the rdil fir t. Simplify. Use the Quotient Property of exponents to simplify the frtion under the rdil fir t. Rewrite the rdind using perfet fourth power ftors. 0 8 Simplify. TRY IT : : 8. Simplify: 8 6 x7 x y7 y. TRY IT : : 8. Simplify: x x 0 m m 7 n n. Rememer the Quotient to Power Property? It sid we ould rise frtion to power y rising the numertor nd denomintor to the power seprtely. m = m m, 0 We n use similr property to simplify root of frtion. After removing ll ommon ftors from the numertor nd denomintor, if the frtion is not perfet power of the index, we simplify the numertor nd denomintor seprtely. Quotient Property of Rdil Expressions If n n nd re rel numers, 0, nd for ny integer n then, n = n n nd n n n = EXAMPLE 8. HOW TO SIMPLIFY THE QUOTIENT OF RADICAL EXPRESSIONS Simplify: 7m 96.
24 766 Chpter 8 Roots nd Rdils TRY IT : : 8. Simplify: p 9. TRY IT : : 8. Simplify: 8x 00. HOW TO : : SIMPLIFY A SQUARE ROOT USING THE QUOTIENT PROPERTY. Step. Step. Step. Simplify the frtion in the rdind, if possile. Use the Quotient Property to rewrite the rdil s the quotient of two rdils. Simplify the rdils in the numertor nd the denomintor. EXAMPLE 8. Simplify: x x7 y y 8x0 y 8. We nnot simplify the frtion in the rdind. Rewrite using the Quotient Property. Simplify the rdils in the numertor nd the denomintor. x y x y 9x x y Simplify. x x y This OpenStx ook is ville for free t
25 Chpter 8 Roots nd Rdils 767 x 7 y The frtion in the rdind nnot e simplified. Use he Quotient Property to write s two rdils. Rewrite eh rdind s produt using perfet ue ftors. Rewrite the numertor s the produt of two rdils. x 7 y 8x 6 x y x y x Simplify. x x y The frtion in the rdind nnot e simplified Use the Quotient Property to write s two rdils. Rewrite eh rdind s produt using perfet fourth power ftors. Rewrite the numertor s the produt of two rdils. 8x 0 y 8 8x 0 y 8 6x 8 x y 8 x y x Simplify. x x y TRY IT : : 8. Simplify: 80m 080 n 6 d 6 80x0 y. TRY IT : : 8.6 Simplify: u7 0r v 8 s 6 6m n. Be sure to simplify the frtion in the rdind first, if possile. EXAMPLE 8. Simplify: 8p q 7 6x y 7 pq x y
26 768 Chpter 8 Roots nd Rdils 8p q 7 pq Simplify the frtion in the rdind, if possile. Rewrite using the Quotient Property. Simplify the rdils in the numertor nd the denomintor. 9p q 6 9p q 6 9p q q Simplify. p q q Simplify the frtion in the rdind, if possile. Rewrite using the Quotient Property. Simplify the rdils in the numertor nd the denomintor. Simplify. 6x y 7 x y 8x y 7 8x y 7 8x y 7 xy y y This OpenStx ook is ville for free t
27 Chpter 8 Roots nd Rdils Simplify the frtion in the rdind, if possile. 6 Rewrite using the Quotient Property. 6 Simplify the rdils in the numertor nd the denomintor. Simplify. 6 TRY IT : : 8.7 Simplify: 0x y 6x y 7 7x y x y TRY IT : : 8.8 Simplify: 8m7 n x7 y 00m n 8 0x y In the next exmple, there is nothing to simplify in the denomintors. Sine the index on the rdils is the sme, we n use the Quotient Property gin, to omine them into one rdil. We will then look to see if we n simplify the expression. EXAMPLE 8. Simplify: x7. x The denomintor nnot e simplified, s use the Quotient Property to write s one rdil Simplify the frtion under the rdil. 6 6 Simplify.
28 770 Chpter 8 Roots nd Rdils The denomintor nnot e simplified, s use the Quotient Property to write s one rdil. Simplify the frtion under the rdil. Rewrite the rdind s produt using perfet ue ftors. Rewrite the rdil s the produt of two rdils ( ) ( ) Simplify. The denomintor nnot e simplified, s use the Quotient Property to write s one rdil. 96x 7 x 96x 7 x Simplify the frtion under the rdil. x Rewrite the rdind s produt using perfet fourth power ftors. Rewrite the rdil s the produt of two rdils. Simplify. 6x x (x) x x x TRY IT : : 8.9 Simplify: 98z z 00 86m. m TRY IT : : 8.0 Simplify: 8m9 m 9 n7. n MEDIA : : Aess these online resoures for dditionl instrution nd prtie with simplifying rdil expressions. Simplifying Squre Root nd Cue Root with Vriles ( Express Rdil in Simplified FormSqure nd Cue Roots with Vriles nd Exponents ( Simplifying Cue Roots ( This OpenStx ook is ville for free t
29 Chpter 8 Roots nd Rdils EXERCISES Prtie Mkes Perfet Use the Produt Property to Simplify Rdil Expressions In the following exerises, use the Produt Property to simplify rdil expressions In the following exerises, simplify using solute vlue signs s needed. 67. y r s m u 7 6 v 69. n q 8 8 n r p 8 m 7. m 0m0 60n 8 7. r 08x 8y n p q s m 7 n 8x 6 y 7 x y r s 80x 7 y 6 80x 8 y q r 7 m 9 n m 9 n 8p 7 q 8 6 d
30 77 Chpter 8 Roots nd Rdils Use the Quotient Property to Simplify Rdil Expressions In the following exerises, use the Quotient Property to simplify squre roots x0 p x 6 p q7 q 9. p0 p 0 d d 7 8 m m 9. y y 8 u u 6 v0 v 9. q8 q r r x y m n r s q r r9 s d 0. 7x y 6 96r s 6 8u7 v 0. 8p7 q 8s8 t 6p q 06. r s 0 6u0 v 79 d x y 8x y x6 y 9 0x y r6 s 8 8rs x8 y 8x y m9 n 6mn 09. 7p q 08p q 6 d 7 0 d 6 m9 n 7 8m n 0. 0r s 8r s 6 m9 n 7 7m n 8m n 8 6m n This OpenStx ook is ville for free t
31 Chpter 8 Roots nd Rdils 77. p9 q 6 8x8 x. 80q q 6 80m7 m. 0m7 m 0 86y9 y. 7n n r0 r Writing Exerises. Explin why x = x. Then explin why x 6 = x Explin why is not equl to Explin how you know tht x 0 = x. 8. Explin why 6 is. is not rel numer ut 6 Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. After reviewing this heklist, wht will you do to eome onfident for ll ojetives?
32 77 Chpter 8 Roots nd Rdils 8. Simplify Rtionl Exponents Lerning Ojetives By the end of this setion, you will e le to: n Simplify expressions with m Simplify expressions with n Use the properties of exponents to simplify expressions with rtionl exponents Be Prepred! Before you get strted, tke this rediness quiz.. Add: 7 +. If you missed this prolem, review Exmple.8.. Simplify: x y. If you missed this prolem, review Exmple.8.. Simplify:. If you missed this prolem, review Exmple.. n Simplify Expressions with Rtionl exponents re nother wy of writing expressions with rdils. When we use rtionl exponents, we n pply the properties of exponents to simplify expressions. The Power Property for Exponents sys tht ( m ) n = m n when m nd n re whole numers. Let s ssume we re now not limited to whole numers. Suppose we wnt to find numer p suh tht (8 p ) = 8. We will use the Power Property of Exponents to find the vlue of p. (8 p ) = 8 Multiply the exponents on the left. 8 p = 8 Write the exponent on the right. 8 p = 8 Sine the ses re the sme, the exponents must e equl. p = Solve for p. p = So 8 = 8. But we know lso 8 = 8. Then it must e tht 8 = 8. This sme logi n e used for ny positive integer exponent n to show tht n = n. n Rtionl Exponent If n is rel numer nd n, then n = n The denomintor of the rtionl exponent is the index of the rdil. There will e times when working with expressions will e esier if you use rtionl exponents nd times when it will e esier if you use rdils. In the first few exmples, you ll prtie onverting expressions etween these two nottions. This OpenStx ook is ville for free t
33 Chpter 8 Roots nd Rdils 77 EXAMPLE 8.6 Write s rdil expression: x y z. We wnt to write eh expression in the form n. The denomintor of the rtionl exponent is, so the index of the rdil is. We do not show the index when it is. The denomintor of the exponent is, so the index is. The denomintor of the exponent is, so the index is. x x y y z z TRY IT : : 8. Write s rdil expression: t m r. TRY IT : : 8. Write s rdil expression: 6 z p. In the next exmple, we will write eh rdil using rtionl exponent. It is importnt to use prentheses round the entire expression in the rdind sine the entire expression is rised to the rtionl power. EXAMPLE 8.7 Write with rtionl exponent: y x z. We wnt to write eh rdil in the form n. No index is shown, so it is. The denomintor of the exponent will e. Put prentheses round the entire expression y. y y
34 776 Chpter 8 Roots nd Rdils The index is, so the denomintor of the exponent is. Inlude prentheses x. The index is, so the denomintor of the exponent is. Put prentheses only round the z sine is not under the rdil sign. x (x) z (z) TRY IT : : 8. Write with rtionl exponent: 0m n 6y. TRY IT : : 8. 7 Write with rtionl exponent: k j 8. In the next exmple, you my find it esier to simplify the expressions if you rewrite them s rdils first. EXAMPLE 8.8 Simplify: 6 6. Rewrite s squre root. Simplify. 6 Rewrite s ue root. 6 Reognize 6 is perfet ue. Simplify. 6 Rewrite s fourth root. 6 Reognize 6 is perfet fourth power. Simplify. TRY IT : : 8. Simplify: This OpenStx ook is ville for free t
35 Chpter 8 Roots nd Rdils 777 TRY IT : : 8.6 Simplify: Be reful of the plement of the negtive signs in the next exmple. We will need to use the property n = n in one se. EXAMPLE 8.9 Simplify: ( 6) 6 (6). Rewrite s fourth root. Simplify. ( 6) 6 ( ) No rel solution. The exponent only pplies to the 6. Rewrite s fouth root. 6 6 Rewrite 6 s. Simplify. (6) Rewrite using the property n = n. (6) Rewrite s fourth root. 6 Rewrite 6 s. Simplify. TRY IT : : 8.7 Simplify: ( 6) 6 (6). TRY IT : : 8.8 Simplify: ( 6) 6 (6). m Simplify Expressions with n m We n look t n n in two wys. Rememer the Power Property tells us to multiply the exponents nd so m nd
36 778 Chpter 8 Roots nd Rdils ( m n ) oth equl m n. If we write these expressions in rdil form, we get m n = n m = ( n ) m m nd n = ( m ) n = n m This leds us to the following definition. m Rtionl Exponent n For ny positive integers m nd n, m n = ( n ) m m nd n n = m Whih form do we use to simplify n expression? We usully tke the root first tht wy we keep the numers in the rdind smller, efore rising it to the power indited. EXAMPLE 8.0 Write with rtionl exponent: m We wnt to use n n = m y x. to write eh rdil in the form m n. This OpenStx ook is ville for free t
37 Chpter 8 Roots nd Rdils 779 TRY IT : : 8.9 Write with rtionl exponent: x y m n. TRY IT : : 8.60 Write with rtionl exponent: 7xy z. Rememer tht n = n. The negtive sign in the exponent does not hnge the sign of the expression. EXAMPLE 8. Simplify: 6. m We will rewrite the expression s rdil first using the defintion, n = ( n ) m. This form lets us tke the root first nd so we keep the numers in the rdind smller thn if we used the other form. The power of the rdil is the numertor of the exponent,. The index of the rdil is the denomintor of the exponent,. Simplify. () We will rewrite eh expression first using n = n nd then hnge to rdil form. 6 Rewrite using n = n 6 Chnge to rdil form. The power of the rdil is the numertor of the exponent,. The index is the denomintor of the exponent,. 6 Simplify. 6
38 780 Chpter 8 Roots nd Rdils Rewrite using n = n. Chnge to rdil form. Rewrite the rdind s power. Simplify. TRY IT : : 8.6 Simplify: TRY IT : : 8.6 Simplify: 7 6. EXAMPLE 8. Simplify: ( ) Rewrite in rdil form. Simplify the rdil. () Simplify.. This OpenStx ook is ville for free t
39 Chpter 8 Roots nd Rdils 78 Rewrite using n = n. Rewrite in rdil form. Simplify the rdil. () Simplify. Rewrite in rdil form. There is no rel numer whose squre root is. ( ) Not rel numer. TRY IT : : 8.6 Simplify: 6 6 ( 6). TRY IT : : 8.6 Simplify: 8 8 ( 8). Use the Properties of Exponents to Simplify Expressions with Rtionl Exponents The sme properties of exponents tht we hve lredy used lso pply to rtionl exponents. We will list the Properties of Exponenets here to hve them for referene s we simplify expressions. Properties of Exponents If nd re rel numers nd m nd n re rtionl numers, then Produt Property m n = m + n Power Property ( m ) n = m n Produt to Power () m = m m Quotient Property m n = m n, 0 Zero Exponent Definitio 0 =, 0 Quotient to Power Property m = m m, 0 Negtive Exponent Property n = n, 0 We will pply these properties in the next exmple. EXAMPLE 8. Simplify: x x 6 z 9 x. x
40 78 Chpter 8 Roots nd Rdils The Produt Property tells us tht when we multiply the sme se, we dd the exponents. x 6 x The ses re the sme, so we dd the exponents. Add the frtions. Simplify the exponent. x x x The Power Property tells us tht when we rise power to power, we multiply the exponents. To rise power to power, we multiply the exponents. z 9 z 9 Simplify. z 6 The Quotient Property tells us tht when we divide with the sme se, we sutrt the exponents. x x x x To divide with the sme se, we sutrt the exponents. x Simplify. x TRY IT : : 8.6 Simplify: x 6 x x 6 x. x TRY IT : : 8.66 Simplify: y y 8 m 9 9 d. 6 d Sometimes we need to use more thn one property. In the next exmple, we will use oth the Produt to Power Property nd then the Power Property. This OpenStx ook is ville for free t
41 Chpter 8 Roots nd Rdils 78 EXAMPLE 8. Simplify: 7u m n. First we use the Produt to Power Property. Rewrite 7 s power of. To rise power to power, we multiply the exponents. Simplify. First we use the Produt to Power Property. To rise power to power, we multiply the exponents. 7u (7) m u u u 9u n m n mn TRY IT : : 8.67 Simplify: x x y. TRY IT : : 8.68 Simplify: 8n. We will use oth the Produt Property nd the Quotient Property in the next exmple. EXAMPLE 8.
42 78 Chpter 8 Roots nd Rdils Simplify: x x x 6 6 x y 6 x. 6 y x x x 6 Use the Produt Property in the numertor, dd the exponents. Use the Quotient Property, sutrt the exponents. x x 6 8 x Simplify. x Follow the order of opertions to simplify inside the prenthese first. Use the Quotient Property, sutrt the exponents. Simplify. Use the Produt to Power Property, multiply the exponents. 6 x x y y x 6 6 y 6x y x y TRY IT : : 8.69 Simplify: m m m m 6 n 6 m n. 6 TRY IT : : 8.70 Simplify: u u u 7x x 6 y y 6. This OpenStx ook is ville for free t
43 Chpter 8 Roots nd Rdils 78 MEDIA : : Aess these online resoures for dditionl instrution nd prtie with simplifying rtionl exponents. ReviewRtionl Exponents ( Using Lws of Exponents on Rdils: Properties of Rtionl Exponents ( 7RtExpont)
44 786 Chpter 8 Roots nd Rdils 8. EXERCISES Prtie Mkes Perfet n Simplify expressions with In the following exerises, write s rdil expression. 9. x y z 0. r s t. u v 9 0 w. g 7 h j In the following exerises, write with rtionl exponent x y f 8. r 0 s t. 7 7 d 6 6. x 8 9y 7 z 7. p 8q 6 6r In the following exerises, simplify ( 6) 6 (6). ( 000) 000 (000). ( 8) 8 (8) 6. ( 9) 9 (9) 7. ( 6) 6 (6) 8. ( 6) ( 00) 00 (00) 0. ( ) () This OpenStx ook is ville for free t
45 Chpter 8 Roots nd Rdils 787 m Simplify Expressions with n In the following exerises, write with rtionl exponent.. m 7 y x y. r 7 pq m 7n. u 6x 8 7. v xy z In the following exerises, simplify ( 7) 6. 9 ( 6) 7. 7 ( ) ( 00) ( 9) ( 6) Use the Lws of Exponents to Simplify Expressions with Rtionl Exponents In the following exerises, simplify.. 8 p r 9 r. 6 6 w w. y y x m 8 8 m
46 788 Chpter 8 Roots nd Rdils. q 6 q h 6 n 8 n. 7q 6. 6s 7 6 m n 7. 6 u p q n 9 x y 9. r r r 6 s t s 9 t x y 7 x y 6. 7 m m m 6 m n 9 8 m n Writing Exerises 6. Show two different lgeri methods to simplify. Explin ll your steps. 6. Explin why the expression ( 6) nnot e evluted. Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. Wht does this heklist tell you out your mstery of this setion? Wht steps will you tke to improve? This OpenStx ook is ville for free t
47 Chpter 8 Roots nd Rdils Add, Sutrt, nd Multiply Rdil Expressions Lerning Ojetives By the end of this setion, you will e le to: Add nd sutrt rdil expressions Multiply rdil expressions Use polynomil multiplition to multiply rdil expressions Be Prepred! Before you get strted, tke this rediness quiz.. Add: x + 9x x x +. If you missed this prolem, review Exmple... Simplify: ( + )( ). If you missed this prolem, review Exmple.8.. Simplify: 9 y. If you missed this prolem, review Exmple.. Add nd Sutrt Rdil Expressions Adding rdil expressions with the sme index nd the sme rdind is just like dding like terms. We ll rdils with the sme index nd the sme rdind like rdils to remind us they work the sme s like terms. Like Rdils Like rdils re rdil expressions with the sme index nd the sme rdind. We dd nd sutrt like rdils in the sme wy we dd nd sutrt like terms. We know tht x + 8x is x. Similrly we dd x + 8 x nd the result is x. Think out dding like terms with vriles s you do the next few exmples. When you hve like rdils, you just dd or sutrt the oeffiients. When the rdils re not like, you nnot omine the terms. EXAMPLE 8.6 Simplify: 7 y + y 7 x y. Sine the rdils re like, we sutrt the oeffiient Sine the rdils re like, we dd the oeffiient 7 y + y 9 y 7 x y The indies re the sme ut the rdils re different. These re not like rdils. Sine the rdils re not like, we nnot sutrt them. TRY IT : : 8.7 Simplify: 8 9 x + 7 x x y.
48 790 Chpter 8 Roots nd Rdils TRY IT : : 8.7 Simplify: 9 y + y m m. For rdils to e like, they must hve the sme index nd rdind. When the rdinds ontin more thn one vrile, s long s ll the vriles nd their exponents re identil, the rdinds re the sme. EXAMPLE 8.7 Simplify: n 6 n + n xy + xy xy. n 6 n + n Sine the rdils re like, we omine them. 0 n Simplify. 0 Sine the rdils re like, we omine them. xy + xy xy xy TRY IT : : 8.7 Simplify: 7x 7 7x + 7x xy + xy 7 xy. TRY IT : : 8.7 Simplify: y 7 y + y 6 7mn + 7mn 7mn. Rememer tht we lwys simplify rdils y removing the lrgest ftor from the rdind tht is power of the index. One eh rdil is simplified, we n then deide if they re like rdils. EXAMPLE 8.8 Simplify: Simplify the rdils, when possile. + + Comine the like rdils. Simplify the rdils. 8 7 Comine the like rdils. This OpenStx ook is ville for free t
49 Chpter 8 Roots nd Rdils 79 8 Simplify the rdils. 6 8 Comine the like rdils. TRY IT : : 8.7 Simplify: TRY IT : : 8.76 Simplify: In the next exmple, we will remove oth onstnt nd vrile ftors from the rdils. Now tht we hve prtied tking oth the even nd odd roots of vriles, it is ommon prtie t this point for us to ssume ll vriles re greter thn or equl to zero so tht solute vlues re not needed. We will use this ssumption thoughout the rest of this hpter. EXAMPLE 8.9 Simplify: 9 0m 6 8m n 6n. 9 0m 6 8m Simplify the rdils. 9 m 6 6m The rdils re not like nd so nnot e omined. Simplify the rdils. 7n 9 m 6 m m m n 6n n 8n n n n n Comine the like rdils. n n n TRY IT : : 8.77 Simplify: m 7 0m 7 x 7 0x 7. TRY IT : : 8.78 Simplify: 7p 8p 6y n. Multiply Rdil Expressions We hve used the Produt Property of Roots to simplify squre roots y removing the perfet squre ftors. We n use the Produt Property of Roots in reverse to multiply squre roots. Rememer, we ssume ll vriles re greter thn or equl to zero. We will rewrite the Produt Property of Roots so we see oth wys together.
50 79 Chpter 8 Roots nd Rdils Produt Property of Roots For ny rel numers, n n nd, nd for ny integer n n = n n nd n n n = When we multiply two rdils they must hve the sme index. One we multiply the rdils, we then look for ftors tht re power of the index nd simplify the rdil whenever possile. Multiplying rdils with oeffiients is muh like multiplying vriles with oeffiients. To multiply x y we multiply the oeffiients together nd then the vriles. The result is xy. Keep this in mind s you do these exmples. EXAMPLE 8.0 Simplify: Multiply using the Produt Property. 8 0 Simplify the rdil. 8 Simplify Multiply using the Produt Property. 0 Simplify the rdil. 0 8 Simplify. 0 0 TRY IT : : 8.79 Simplify: TRY IT : : 8.80 Simplify: We follow the sme proedures when there re vriles in the rdinds. EXAMPLE 8. Simplify: 0 6p p 0y 8y. This OpenStx ook is ville for free t
51 Chpter 8 Roots nd Rdils p p Multiply. 0 8p Simplify the rdil. 0 9p Simplify. 0 p 0p When the rdinds involve lrge numers, it is often dvntgeous to ftor them in order to find the perfet powers. 0y 8y Multiply. 6 7y Simplify the rdil. 6 6y y Simplify. Multiply. 6 y y y y TRY IT : : 8.8 Simplify: 6 6x 8 0x y 8y. TRY IT : : 8.8 Simplify: 6y 0y 9 7. Use Polynomil Multiplition to Multiply Rdil Expressions In the next few exmples, we will use the Distriutive Property to multiply expressions with rdils. First we will distriute nd then simplify the rdils when possile. EXAMPLE 8. Simplify: Multiply Simplify. + 6 Simplify. + 6 Comine like rdils Distriute. 9 6 Simplify Simplify. 9 6
52 79 Chpter 8 Roots nd Rdils TRY IT : : 8.8 Simplify: TRY IT : : 8.8 Simplify: When we worked with polynomils, we multiplied inomils y inomils. Rememer, this gve us four produts efore we omined ny like terms. To e sure to get ll four produts, we orgnized our work usully y the FOIL method. EXAMPLE 8. Simplify: ( 7)( 7) x x +. ( 7)( 7) Multiply Simplify Comine like terms. 0 7 x x + Multiply. x + x x 8 Comine like terms. x + x 8 TRY IT : : 8.8 Simplify: (6 7) + 7 x x. TRY IT : : 8.86 Simplify: ( ) x + x +. EXAMPLE 8. Simplify: +. + Multiply Simplify Comine like terms. + 0 TRY IT : : 8.87 Simplify: TRY IT : : 8.88 Simplify: Reognizing some speil produts mde our work esier when we multiplied inomils erlier. This is true when we multiply rdils, too. The speil produt formuls we used re shown here. This OpenStx ook is ville for free t
53 Chpter 8 Roots nd Rdils 79 Speil Produts Binomil Squres Produt of Conjugtes ( + ) = + + ( + )( ) = ( ) = + We will use the speil produt formuls in the next few exmples. We will strt with the Produt of Binomil Squres Pttern. EXAMPLE 8. Simplify: +. Be sure to inlude the term when squring inomil. Multiply, using the Produt of Binomil Squres Pttern. Simplify. Comine like terms. Multiply, using the Produt of Binomil Squres Pttern. Simplify. Comine like terms. TRY IT : : 8.89 Simplify: TRY IT : : 8.90 Simplify: In the next exmple, we will use the Produt of Conjugtes Pttern. Notie tht the finl produt hs no rdil. EXAMPLE 8.6 Simplify: +.
54 796 Chpter 8 Roots nd Rdils Multiply, using the Produt of Conjugtes Pttern. Simplify. TRY IT : : 8.9 Simplify: + TRY IT : : 8.9 Simplify: + 7 ( 7). MEDIA : : Aess these online resoures for dditionl instrution nd prtie with dding, sutrting, nd multiplying rdil expressions. Multiplying Adding Sutrting Rdils ( Multiplying Speil Produts: Squre Binomils Contining Squre Roots ( 7Rdils) Multiplying Conjugtes ( This OpenStx ook is ville for free t
55 Chpter 8 Roots nd Rdils EXERCISES Prtie Mkes Perfet Add nd Sutrt Rdil Expressions In the following exerises, simplify m + m 8 m n p + p x x z + z m m n + n d + d 9 d pq pq + pq 7. d + 8 d d rs 9 rs + rs p 6 0p q 6 q r 0 + r d 9 d 9 s 6 + s y + y y 8. 7y + 8y 8 00y Multiply Rdil Expressions In the following exerises, simplify. 8. ( )( 8) ( )( 0) ( 6)( ) 8 9
56 798 Chpter 8 Roots nd Rdils 86. ( 7)( ) z 9z x 8x 88. x 7 8x z z 8 8y y 90. k k Use Polynomil Multiplition to Multiply Rdil Expressions In the following exerises, multiply x x ( 7)( 7) x x x + 6 x x + x ( ) x + x 6. x + x Mixed Prtie k 6k This OpenStx ook is ville for free t
57 Chpter 8 Roots nd Rdils / q 6 q x 6x. 9. ( 7) Writing Exerises. Explin the when rdil expression is in simplest form.. Explin why ( n) is lwys nonnegtive, for n 0. Explin why ( n) is lwys nonpositive, for n 0.. Explin the proess for determining whether two rdils re like or unlike. Mke sure your nswer mkes sense for rdils ontining oth numers nd vriles.. Use the inomil squre pttern to simplify +. Explin ll your steps. Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. On sle of 0, how would you rte your mstery of this setion in light of your responses on the heklist? How n you improve this?
58 800 Chpter 8 Roots nd Rdils 8. Divide Rdil Expressions Lerning Ojetives By the end of this setion, you will e le to: Divide rdil expressions Rtionlize one term denomintor Rtionlize two term denomintor Be Prepred! Before you get strted, tke this rediness quiz.. Simplify: 0 8. If you missed this prolem, review Exmple... Simplify: x x. If you missed this prolem, review Exmple... Multiply: (7 + x)(7 x). If you missed this prolem, review Exmple.. Divide Rdil Expressions We hve used the Quotient Property of Rdil Expressions to simplify roots of frtions. We will need to use this property in reverse to simplify frtion with rdils. We give the Quotient Property of Rdil Expressions gin for esy referene. Rememer, we ssume ll vriles re greter thn or equl to zero so tht no solute vlue rs re needed. Quotient Property of Rdil Expressions If n n nd re rel numers, 0, nd for ny integer n then, n = n n nd n n n = We will use the Quotient Property of Rdil Expressions when the frtion we strt with is the quotient of two rdils, nd neither rdind is perfet power of the index. When we write the frtion in single rdil, we my find ommon ftors in the numertor nd denomintor. EXAMPLE 8.7 Simplify: 7x 6x x. x This OpenStx ook is ville for free t
59 Chpter 8 Roots nd Rdils 80 7x 6x Rewrite using the quotient property, n n n = 7x. 6x Remove ommon ftors. 8 x x 8 9 x Simplify. x 9 Simplify the rdil. x x x Rewrite using the quotient property, n n n = x. x Simplify the frtion under the rdil. 8 x Simplify the rdil. x TRY IT : : 8.9 Simplify: 0s 8s 6. 7 TRY IT : : 8.9 Simplify: 7q 08q 7. 9 EXAMPLE 8.8 Simplify: 78 0m n. m n 7 8 Rewrite using the quotient property. 7 8 Remove ommon ftors in the frtion. 9 Simplify the rdil. 7
60 80 Chpter 8 Roots nd Rdils 0m n m n Rewrite using the quotient property. 0m n m n Simplify the frtion under the rdil. m Simplify the rdil. n 6 m n TRY IT : : 8.9 Simplify: 6x0 y 8x y. x 6 y 6 x y TRY IT : : 8.96 Simplify: 00m n 7 m n 8pq. p q EXAMPLE 8.9 Simplify: x y. x y x y x y Rewrite using the quotient property. x y x y Remove ommon ftors in the frtion. 8x y Rewrite the rdind s produt using the lrgest perfet squre ftor. 9x y x Rewrite the rdil s the produt of two rdils. 9x y x Simplify. xy x TRY IT : : 8.97 Simplify: 6x y xy. TRY IT : : 8.98 Simplify: 96. Rtionlize One Term Denomintor Before the lultor eme tool of everydy life, pproximting the vlue of frtion with rdil in the denomintor ws very umersome proess! For this reson, proess lled rtionlizing the denomintor ws developed. A frtion with rdil in the denomintor is onverted to n equivlent frtion whose denomintor is n integer. Squre roots of numers tht re not perfet squres re irrtionl numers. When we rtionlize the denomintor, we write n equivlent frtion with This OpenStx ook is ville for free t
61 Chpter 8 Roots nd Rdils 80 rtionl numer in the denomintor. This proess is still used tody, nd is useful in other res of mthemtis, too. Rtionlizing the Denomintor Rtionlizing the denomintor is the proess of onverting frtion with rdil in the denomintor to n equivlent frtion whose denomintor is n integer. Even though we hve lultors ville nerly everywhere, frtion with rdil in the denomintor still must e rtionlized. It is not onsidered simplified if the denomintor ontins rdil. Similrly, rdil expression is not onsidered simplified if the rdind ontins frtion. Simplified Rdil Expressions A rdil expression is onsidered simplified if there re no ftors in the rdind hve perfet powers of the index no frtions in the rdind no rdils in the denomintor of frtion To rtionlize denomintor with squre root, we use the property tht ( ) =. If we squre n irrtionl squre root, we get rtionl numer. We will use this property to rtionlize the denomintor in the next exmple. EXAMPLE 8.0 Simplify: 0 6x. To rtionlize denomintor with one term, we n multiply squre root y itself. To keep the frtion equivlent, we multiply oth the numertor nd denomintor y the sme ftor. Multiply oth the numertor nd denomintor y. Simplify. We lwys simplify the rdil in the denomintor first, efore we rtionlize it. This wy the numers sty smller nd esier to work with. The frtion is not perfet squre, so rewrite using the Quotient Property. Simplify the denomintor. Multiply the numertor nd denomintor y.
62 80 Chpter 8 Roots nd Rdils Simplify. Simplify. Multiply the numertor nd denomintor y 6x. Simplify. Simplify. TRY IT : : 8.99 Simplify: x. TRY IT : : 8.00 Simplify: x. When we rtionlized squre root, we multiplied the numertor nd denomintor y squre root tht would give us perfet squre under the rdil in the denomintor. When we took the squre root, the denomintor no longer hd rdil. We will follow similr proess to rtionlize higher roots. To rtionlize denomintor with higher index rdil, we multiply the numertor nd denomintor y rdil tht would give us rdind tht is perfet power of the index. When we simplify the new rdil, the denomintor will no longer hve rdil. For exmple, We will use this tehnique in the next exmples. EXAMPLE 8. Simplify 7 6. x This OpenStx ook is ville for free t
63 Chpter 8 Roots nd Rdils 80 To rtionlize denomintor with ue root, we n multiply y ue root tht will give us perfet ue in the rdind in the denomintor. To keep the frtion equivlent, we multiply oth the numertor nd denomintor y the sme ftor. The rdil in the denomintor hs one ftor of 6. Multiply oth the numertor nd denomintor y 6, whih gives us more ftors of 6. Multiply. Notie the rdind in the denomintor hs powers of 6. Simplify the ue root in the denomintor. We lwys simplify the rdil in the denomintor first, efore we rtionlize it. This wy the numers sty smller nd esier to work with. The frtion is not perfet ue, so rewrite using the Quotient Property. Simplify the denomintor. Multiply the numertor nd denomintor y. This will give us ftors of. Simplify. Rememer, Simplify. =.
64 806 Chpter 8 Roots nd Rdils Rewrite the rdind to show the ftors. Multiply the numertor nd denomintor y x. This will get us ftors of nd ftors of x. Simplify. Simplify the rdil in the denomintor. TRY IT : : 8.0 Simplify: 7. 9y TRY IT : : 8.0 Simplify: 0. n EXAMPLE 8. Simplify: 6. 8x To rtionlize denomintor with fourth root, we n multiply y fourth root tht will give us perfet fourth power in the rdind in the denomintor. To keep the frtion equivlent, we multiply oth the numertor nd denomintor y the sme ftor. The rdil in the denomintor hs one ftor of. Multiply oth the numertor nd denomintor y, whih gives us more ftors of. Multiply. Notie the rdind in the denomintor hs powers of. Simplify the fourth root in the denomintor. We lwys simplify the rdil in the denomintor first, efore we rtionlize it. This wy the numers sty smller nd esier to work with. This OpenStx ook is ville for free t
65 Chpter 8 Roots nd Rdils 807 The frtion is not perfet fourth power, so rewrite using the Quotient Property. Rewrite the rdind in the denomintor to show the ftors. Simplify the denomintor. Multiply the numertor nd denomintor y. This will give us ftors of. Simplify. Rememer, Simplify. =. Rewrite the rdind to show the ftors. Multiply the numertor nd denomintor y x. This will get us ftors of nd ftors of x. Simplify. Simplify the rdil in the denomintor. Simplify the frtion. TRY IT : : 8.0 Simplify: 6. x TRY IT : : 8.0 Simplify: 7 8 x Rtionlize Two Term Denomintor When the denomintor of frtion is sum or differene with squre roots, we use the Produt of Conjugtes Pttern to rtionlize the denomintor.
66 808 Chpter 8 Roots nd Rdils ( )( + ) + When we multiply inomil tht inludes squre root y its onjugte, the produt hs no squre roots. EXAMPLE 8. Simplify:. Multiply the numertor nd denomintor y the onjugte of the denomintor. Multiply the onjugtes in the denomintor. Simplify the denomintor. Simplify the denomintor. Simplify. TRY IT : : 8.0 Simplify:. TRY IT : : 8.06 Simplify: 6. Notie we did not distriute the in the nswer of the lst exmple. By leving the result ftored we n see if there re ny ftors tht my e ommon to oth the numertor nd denomintor. EXAMPLE 8. Simplify: u 6. Multiply the numertor nd denomintor y the onjugte of the denomintor. Multiply the onjugtes in the denomintor. This OpenStx ook is ville for free t
67 Chpter 8 Roots nd Rdils 809 Simplify the denomintor. TRY IT : : 8.07 Simplify: x +. TRY IT : : 8.08 Simplify: 0 y. Be reful of the signs when multiplying. The numertor nd denomintor look very similr when you multiply y the onjugte. EXAMPLE 8. Simplify: x + 7 x 7. Multiply the numertor nd denomintor y the onjugte of the denomintor. Multiply the onjugtes in the denomintor. Simplify the denomintor. We do not squre the numertor. Leving it in ftored form, we n see there re no ommon ftors to remove from the numertor nd denomintor. TRY IT : : 8.09 Simplify: p + p. TRY IT : : 8.0 Simplify: q 0 q + 0 MEDIA : : Aess these online resoures for dditionl instrution nd prtie with dividing rdil expressions. Rtionlize the Denomintor ( Dividing Rdil Expressions nd Rtionlizing the Denomintor ( 7RtDenom) Simplifying Rdil Expression with Conjugte ( Rtionlize the Denomintor of Rdil Expression (
68 80 Chpter 8 Roots nd Rdils 8. EXERCISES Prtie Mkes Perfet Divide Squre Roots In the following exerises, simplify m 98m y y 9. 7r 08r 7 x7 8x n7 n 0. 96q 8q y 6y 6m m. 08p q 6 p q 6. 98rs0 7y z r s y z. 0mn m 7 n 6x y x y. 80 d d x y xy x y 6 x y Rtionlize One Term Denomintor In the following exerises, rtionlize the denomintor x y p q x y x x This OpenStx ook is ville for free t
69 Chpter 8 Roots nd Rdils 8 Rtionlize Two Term Denomintor In the following exerises, simplify m 76. n x y r + r 80. s 6 s x + 8 x 8 8. m m + Writing Exerises 8. Simplify 7 nd explin ll your steps. 8. Explin wht is ment y the word rtionlize in the phrse, rtionlize denomintor. Simplify 7 nd explin ll your steps. Why re the two methods of simplifying squre roots different? 8. Explin why multiplying x y its onjugte results in n expression with no rdils. 86. Explin why multiplying 7 x y x x does not rtionlize the denomintor. Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. After looking t the heklist, do you think you re wellprepred for the next setion? Why or why not?
70 8 Chpter 8 Roots nd Rdils 8.6 Solve Rdil Equtions Lerning Ojetives By the end of this setion, you will e le to: Solve rdil equtions Solve rdil equtions with two rdils Use rdils in pplitions Be Prepred! Before you get strted, tke this rediness quiz.. Simplify: y. If you missed this prolem, review Exmple... Solve: x = 0. If you missed this prolem, review Exmple... Solve n 6n + 8 = 0. If you missed this prolem, review Exmple 6.. Solve Rdil Equtions In this setion we will solve equtions tht hve vrile in the rdind of rdil expression. An eqution of this type is lled rdil eqution. Rdil Eqution An eqution in whih vrile is in the rdind of rdil expression is lled rdil eqution. As usul, when solving these equtions, wht we do to one side of n eqution we must do to the other side s well. One we isolte the rdil, our strtegy will e to rise oth sides of the eqution to the power of the index. This will eliminte the rdil. Solving rdil equtions ontining n even index y rising oth sides to the power of the index my introdue n lgeri solution tht would not e solution to the originl rdil eqution. Agin, we ll this n extrneous solution s we did when we solved rtionl equtions. In the next exmple, we will see how to solve rdil eqution. Our strtegy is sed on rising rdil with index n to the n th power. This will eliminte the rdil. For 0, ( n ) n =. EXAMPLE 8.6 HOW TO SOLVE A RADICAL EQUATION Solve: n 9 = 0. This OpenStx ook is ville for free t
71 Chpter 8 Roots nd Rdils 8 TRY IT : : 8. Solve: m + = 0. TRY IT : : 8. Solve: 0z + = 0. HOW TO : : SOLVE A RADICAL EQUATION WITH ONE RADICAL. Step. Step. Step. Step. Isolte the rdil on one side of the eqution. Rise oth sides of the eqution to the power of the index. Solve the new eqution. Chek the nswer in the originl eqution. When we use rdil sign, it indites the prinipl or positive root. If n eqution hs rdil with n even index equl to negtive numer, tht eqution will hve no solution. EXAMPLE 8.7 Solve: 9k + = 0. To isolte the rdil, sutrt to oth sides. Simplify. Beuse the squre root is equl to negtive numer, the eqution hs no solution. TRY IT : : 8. Solve: r + = 0. TRY IT : : 8. Solve: 7s + = 0. If one side of n eqution with squre root is inomil, we use the Produt of Binomil Squres Pttern when we squre it.
72 8 Chpter 8 Roots nd Rdils Binomil Squres ( + ) = + + ( ) = + Don t forget the middle term! EXAMPLE 8.8 Solve: p + = p. To isolte the rdil, sutrt from oth sides. Simplify. Squre oth sides of the eqution. Simplify, using the Produt of Binomil Squres Pttern on the right. Then solve the new eqution. It is qudrti eqution, so get zero on one side. Ftor the right side. Use the Zero Produt Property. Solve eh eqution. Chek the nswers. The solutions re p =, p =. TRY IT : : 8. Solve: x + = x. TRY IT : : 8.6 Solve: y + = y. When the index of the rdil is, we ue oth sides to remove the rdil. EXAMPLE 8.9 Solve: x =. = This OpenStx ook is ville for free t
73 Chpter 8 Roots nd Rdils 8 x = To isolte the rdil, sutrt 8 from oth sides. Cue oth sides of the eqution. Simplify. Solve the eqution. x + = x + = ( ) x + = 6 x = 6 x = Chek the nswer. The solution is x =. TRY IT : : 8.7 Solve: x + 8 = TRY IT : : 8.8 Solve: 6x 0 + = Sometimes n eqution will ontin rtionl exponents insted of rdil. We use the sme tehniques to solve the eqution s when we hve rdil. We rise eh side of the eqution to the power of the denomintor of the rtionl exponent. Sine ( m ) n = m n, we hve for exmple, x = x, x = x Rememer, x = x nd x = x. EXAMPLE 8.60 Solve: (x ) + =. (x ) + =
74 86 Chpter 8 Roots nd Rdils To isolte the term with the rtionl exponent, sutrt from oth sides. (x ) = Rise eh side of the eqution to the fourth power. (x ) = () Simplify. x = 6 Solve the eqution. x = 8 x = 6 Chek the nswer. The solution is x = 6. TRY IT : : 8.9 Solve: (9x + 9) =. TRY IT : : 8.0 Solve: (x 8) + = 7. Sometimes the solution of rdil eqution results in two lgeri solutions, ut one of them my e n extrneous solution! EXAMPLE 8.6 Solve: r + r + = 0. r + r + = 0 Isolte the rdil. r + = r Squre oth sides of the eqution. r + = (r ) Simplify nd then solve the eqution r + = r r + It is qudrti eqution, so get zero on one side. 0 = r r Ftor the right side. 0 = r(r ) Use the Zero Produt Property. 0 = r 0 = r This OpenStx ook is ville for free t
75 Chpter 8 Roots nd Rdils 87 Solve the eqution. r = 0 r = Chek your nswer. The solution is r =. r = 0 is n extrneous solution. TRY IT : : 8. Solve: m + 9 m + = 0. TRY IT : : 8. Solve: n + n + = 0. When there is oeffiient in front of the rdil, we must rise it to the power of the index, too. EXAMPLE 8.6 Solve: x 8 =. x 8 = Isolte the rdil term. x = Isolte the rdil y dividing oth sides y. x = Squre oth sides of the eqution. x = () Simplify, then solve the new eqution. x = 6 x = Solve the eqution. x = 7 Chek the nswer. The solution is x = 7. TRY IT : : 8. Solve: + 6 = 6.
76 88 Chpter 8 Roots nd Rdils TRY IT : : 8. Solve: + = 0. Solve Rdil Equtions with Two Rdils If the rdil eqution hs two rdils, we strt out y isolting one of them. It often works out esiest to isolte the more omplited rdil first. In the next exmple, when one rdil is isolted, the seond rdil is lso isolted. EXAMPLE 8.6 Solve: x = x +. The rdil terms re isolted. x Sine the index is, ue oth sides of the eqution. x = x + = x + Simplify, then solve the new eqution. x = x + x = x = The solution is x =. Chek the nswer. We leve it to you to show tht heks! TRY IT : : 8. Solve: x = x +. TRY IT : : 8.6 Solve: 7x + = x. Sometimes fter rising oth sides of n eqution to power, we still hve vrile inside rdil. When tht hppens, we repet Step nd Step of our proedure. We isolte the rdil nd rise oth sides of the eqution to the power of the index gin. EXAMPLE 8.6 HOW TO SOLVE A RADICAL EQUATION Solve: m + = m + 9. This OpenStx ook is ville for free t
77 Chpter 8 Roots nd Rdils 89 TRY IT : : 8.7 Solve: x = x. TRY IT : : 8.8 Solve: x + = x + 6. We summrize the steps here. We hve djusted our previous steps to inlude more thn one rdil in the eqution This proedure will now work for ny rdil equtions. HOW TO : : SOLVE A RADICAL EQUATION. Step. Step. Step. Step. Isolte one of the rdil terms on one side of the eqution. Rise oth sides of the eqution to the power of the index. Are there ny more rdils? If yes, repet Step nd Step gin. If no, solve the new eqution. Chek the nswer in the originl eqution. Be reful s you squre inomils in the next exmple. Rememer the pttern is ( + ) = + + or ( ) = +. EXAMPLE 8.6 Solve: q + = q +. The rdil on the right is isolted. Squre oth sides. Simplify. There is still rdil in the eqution so we must repet the previous steps. Isolte the rdil. Squre oth sides. It would not help to divide oth sides y 6. Rememer to squre oth the 6 nd the q. Simplify, then solve the new eqution.
78 80 Chpter 8 Roots nd Rdils Distriute. It is qudrti eqution, so get zero on one side. Ftor the right side. Use the Zero Produt Property. The heks re left to you. The solutions re q = 6 nd q =. TRY IT : : 8.9 Solve: x + = x + 6 TRY IT : : 8.0 Solve: x + = x + Use Rdils in Applitions As you progress through your ollege ourses, you ll enounter formuls tht inlude rdils in mny disiplines. We will modify our Prolem Solving Strtegy for Geometry Applitions slightly to give us pln for solving pplitions with formuls from ny disipline. HOW TO : : USE A PROBLEM SOLVING STRATEGY FOR APPLICATIONS WITH FORMULAS. Step. Step. Step. Step. Step. Step 6. Step 7. Red the prolem nd mke sure ll the words nd ides re understood. When pproprite, drw figure nd lel it with the given informtion. Identify wht we re looking for. Nme wht we re looking for y hoosing vrile to represent it. Trnslte into n eqution y writing the pproprite formul or model for the sitution. Sustitute in the given informtion. Solve the eqution using good lger tehniques. Chek the nswer in the prolem nd mke sure it mkes sense. Answer the question with omplete sentene. One pplition of rdils hs to do with the effet of grvity on flling ojets. The formul llows us to determine how long it will tke fllen ojet to hit the gound. Flling Ojets On Erth, if n ojet is dropped from height of h feet, the time in seonds it will tke to reh the ground is found y using the formul t = h. For exmple, if n ojet is dropped from height of 6 feet, we n find the time it tkes to reh the ground y sustituting h = 6 into the formul. This OpenStx ook is ville for free t
79 Chpter 8 Roots nd Rdils 8 Tke the squre root of 6. Simplify the frtion. It would tke seonds for n ojet dropped from height of 6 feet to reh the ground. EXAMPLE 8.66 Mriss dropped her sunglsses from ridge 00 feet ove river. Use the formul t = it took for the sunglsses to reh the river. h to find how mny seonds Step. Red the prolem. Step. Identify wht we re looking for. Step. Nme wht we re looking. the time it tkes for the sunglsses to reh the river Let t = time. Step. Trnslte into n eqution y writing the pproprite formul. Sustitute in the given informtion. Step. Solve the eqution. Step 6. Chek the nswer in the prolem nd mke sure it mkes sense. Does seonds seem like resonle length of time? Step 7. Answer the question. Yes. It will tke seonds for the sunglsses to reh the river. TRY IT : : 8. A heliopter dropped resue pkge from height of,96 feet. Use the formul t = seonds it took for the pkge to reh the ground. h to find how mny
80 8 Chpter 8 Roots nd Rdils TRY IT : : 8. A window wsher dropped squeegee from pltform 96 feet ove the sidewlk Use the formul t = find how mny seonds it took for the squeegee to reh the sidewlk. h to Polie offiers investigting r idents mesure the length of the skid mrks on the pvement. Then they use squre roots to determine the speed, in miles per hour, r ws going efore pplying the rkes. Skid Mrks nd Speed of Cr If the length of the skid mrks is d feet, then the speed, s, of the r efore the rkes were pplied n e found y using the formul s = d EXAMPLE 8.67 After r ident, the skid mrks for one r mesured 90 feet. Use the formul s = efore the rkes were pplied. Round your nswer to the nerest tenth. d to find the speed of the r Step. Red the prolem Step. Identify wht we re looking for. Step. Nme wht were looking for, the speed of r Let s = the speed. Step. Trnslte into n eqution y writing the pproprite formul. Sustitute in the given informtion. Step. Solve the eqution. Round to deiml ple. The speed of the r efore the rkes were pplied ws 67. miles per hour. TRY IT : : 8. An ident investigtor mesured the skid mrks of the r. The length of the skid mrks ws 76 feet. Use the formul s = d to find the speed of the r efore the rkes were pplied. Round your nswer to the nerest tenth. TRY IT : : 8. The skid mrks of vehile involved in n ident were feet long. Use the formul s = speed of the vehile efore the rkes were pplied. Round your nswer to the nerest tenth. d to find the This OpenStx ook is ville for free t
81 Chpter 8 Roots nd Rdils 8 MEDIA : : Aess these online resoures for dditionl instrution nd prtie with solving rdil equtions. Solving n Eqution Involving Single Rdil ( Solving Equtions with Rdils nd Rtionl Exponents ( Solving Rdil Equtions ( Solve Rdil Equtions ( Rdil Eqution Applition (
82 8 Chpter 8 Roots nd Rdils 8.6 EXERCISES Prtie Mkes Perfet Solve Rdil Equtions In the following exerises, solve. 87. x 6 = x = x + = 90. y = 9. x = 9. x 9. m = 0 9. n = v 0 = u + = m + + = n + + = 8 = 99. u + = v + = 0 0. u = u 0. v = v 0. r = r 0. s 8 = s x + = 06. x + = 07. x + = 08. 9x = 09. (6x + ) = 0. (x ) + = 6. (8x + ) + =. (x ) + 8 =. (x ) =. (x ) + 7 = 9. x + x + = 0 6. y + y + = 0 7. z + 00 z = 0 8. w + w = 9. x 0 = 7 0. x + 8 = 0. 8r + 8 =. 7y + 0 = 8 Solve Rdil Equtions with Two Rdils In the following exerises, solve.. u + 7 = u +. v + = v r = r = x = x x = x + 9. x + 9x 8 = x + x 0. x x + 8 = x x 6. + = +. r + 6 = r + 8. u + = u +. x + = x +. + = 6. = d 0 d 7. x + = + x 8. x + = + x 9. x x = 0. x + x = This OpenStx ook is ville for free t
83 Chpter 8 Roots nd Rdils 8. x + 7 x =. x + x = Use Rdils in Applitions In the following exerises, solve. Round pproximtions to one deiml ple.. Lndsping Reed wnts to hve squre grden plot in his kyrd. He hs enough ompost to over n re of 7 squre feet. Use the formul s = A to find the length of eh side of his grden. Round your nswer to the nerest tenth of foot.. Lndsping Vine wnts to mke squre ptio in his yrd. He hs enough onrete to pve n re of 0 squre feet. Use the formul s = A to find the length of eh side of his ptio. Round your nswer to the nerest tenth of foot.. Grvity A hng glider dropped his ell phone from height of 0 feet. Use the formul t = h to find how mny seonds it took for the ell phone to reh the ground. 6. Grvity A onstrution worker dropped hmmer while uilding the Grnd Cnyon skywlk, 000 feet ove the Colordo River. Use the formul t = h to find how mny seonds it took for the hmmer to reh the river. Writing Exerises 7. Aident investigtion The skid mrks for r involved in n ident mesured 6 feet. Use the formul s = d to find the speed of the r efore the rkes were pplied. Round your nswer to the nerest tenth. 8. Aident investigtion An ident investigtor mesured the skid mrks of one of the vehiles involved in n ident. The length of the skid mrks ws 7 feet. Use the formul s = d to find the speed of the vehile efore the rkes were pplied. Round your nswer to the nerest tenth. 9. Explin why n eqution of the form x + = 0 hs no solution. 0. Solve the eqution r + r + = 0. Explin why one of the solutions tht ws found ws not tully solution to the eqution. Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. After reviewing this heklist, wht will you do to eome onfident for ll ojetives?
84 86 Chpter 8 Roots nd Rdils 8.7 Use Rdils in Funtions Lerning Ojetives By the end of this setion, you will e le to: Evlute rdil funtion Find the domin of rdil funtion Grph rdil funtions Be Prepred! Before you get strted, tke this rediness quiz.. Solve: x 0. If you missed this prolem, review Exmple.0.. For f (x) = x, evlute f (), f ( ), f (0). If you missed this prolem, review Exmple.8.. Grph f (x) = x. Stte the domin nd rnge of the funtion in intervl nottion. If you missed this prolem, review Exmple.6. Evlute Rdil Funtion In this setion we will extend our previous work with funtions to inlude rdils. If funtion is defined y rdil expression, we ll it rdil funtion. The squre root funtion is f (x) = x. The ue root funtion is f (x) = x. Rdil Funtion A rdil funtion is funtion tht is defined y rdil expression. To evlute rdil funtion, we find the vlue of f(x) for given vlue of x just s we did in our previous work with funtions. EXAMPLE 8.68 For the funtion f (x) = x, find f () f ( ). f (x) = x To evlute f (), sustitute for x. f () = Simplify. f () = 9 Tke the squre root. f () = f (x) = x To evlute f ( ), sustitute for x. f ( ) = ( ) Simplify. f ( ) = Sine the squre root of negtive numer is not rel numer, the funtion does not hve vlue t x =. TRY IT : : 8. For the funtion f (x) = x, find f (6) f (0). This OpenStx ook is ville for free t
85 Chpter 8 Roots nd Rdils 87 TRY IT : : 8.6 For the funtion g(x) = x +, find g() g( ). We follow the sme proedure to evlute ue roots. EXAMPLE 8.69 For the funtion g(x) = x 6, find g() g( ). g(x) = x 6 To evlute g(), sustitute for x. g() = 6 Simplify. g() = 8 Tke the ue root. g() = g(x) = x 6 To evlute g( ), sustitute for x. g( ) = 6 Simplify. g( ) = 8 Tke the ue root. g( ) = TRY IT : : 8.7 For the funtion g(x) = x, find g() g(). TRY IT : : 8.8 For the funtion h(x) = x, find h() h( ). The next exmple hs fourth roots. EXAMPLE 8.70 For the funtion f (x) = x, find f () f ( ) f (x) = x To evlute f (), sustitute for x. f () = Simplify. f () = 6 Tke the fourth root. f () = f (x) = x To evlute f ( ), sustitute for x. f ( ) = ( ) Simplify. f ( ) = 6 Sine the fourth root of negtive numer is not rel numer, the funtion does not hve vlue t x =.
86 88 Chpter 8 Roots nd Rdils TRY IT : : 8.9 For the funtion f (x) = x +, find f () f ( ). TRY IT : : 8.0 For the funtion g(x) = x +, find g(6) g(). Find the Domin of Rdil Funtion To find the domin nd rnge of rdil funtions, we use our properties of rdils. For rdil with n even index, we sid the rdind hd to e greter thn or equl to zero s even roots of negtive numers re not rel numers. For n odd index, the rdind n e ny rel numer. We restte the properties here for referene. Properties of n When n is n even numer nd: 0, then n is rel numer. < 0, then n When n is n odd numer, is not rel numer. n is rel numer for ll vlues of. So, to find the domin of rdil funtion with even index, we set the rdind to e greter thn or equl to zero. For n odd index rdil, the rdind n e ny rel numer. Domin of Rdil Funtion When the index of the rdil is even, the rdind must e greter thn or equl to zero. When the index of the rdil is odd, the rdind n e ny rel numer. EXAMPLE 8.7 Find the domin of the funtion, f (x) = x. Write the domin in intervl nottion. Sine the funtion, f (x) = x hs rdil with n index of, whih is even, we know the rdind must e greter thn or equl to 0. We set the rdind to e greter thn or equl to 0 nd then solve to find the domin. x 0 Solve. x x The domin of f (x) = x is ll vlues x nd we write it in intervl nottion s,. TRY IT : : 8. Find the domin of the funtion, f (x) = 6x. Write the domin in intervl nottion. TRY IT : : 8. Find the domin of the funtion, f (x) = x. Write the domin in intervl nottion. EXAMPLE 8.7 Find the domin of the funtion, g(x) = 6. Write the domin in intervl nottion. x Sine the funtion, g(x) = 6 hs rdil with n index of, whih is even, we know the rdind must e greter x thn or equl to 0. This OpenStx ook is ville for free t
87 Chpter 8 Roots nd Rdils 89 The rdind nnot e zero sine the numertor is not zero. For 6 to e greter thn zero, the denomintor must e positive sine the numertor is positive. We know positive x divided y positive is positive. We set x > 0 nd solve. x > 0 Solve. x > Also, sine the rdind is frtion, we must relize tht the denomintor nnot e zero. We solve x = 0 to find the vlue tht must e eliminted from the domin. x = 0 Solve. x = so x in the domin. Putting this together we get the domin is x > nd we write it s (, ). TRY IT : : 8. Find the domin of the funtion, f (x) =. Write the domin in intervl nottion. x + TRY IT : : 8. Find the domin of the funtion, h(x) = 9. Write the domin in intervl nottion. x The next exmple involves ue root nd so will require different thinking. EXAMPLE 8.7 Find the domin of the funtion, f (x) = x +. Write the domin in intervl nottion. Sine the funtion, f (x) = x + hs rdil with n index of, whih is odd, we know the rdind n e ny rel numer. This tells us the domin is ny rel numer. In intervl nottion, we write (, ). The domin of f (x) = x + is ll rel numers nd we write it in intervl nottion s (, ). TRY IT : : 8. Find the domin of the funtion, f (x) = x. Write the domin in intervl nottion. TRY IT : : 8.6 Find the domin of the funtion, g(x) = x. Write the domin in intervl nottion. Grph Rdil Funtions Before we grph ny rdil funtion, we first find the domin of the funtion. For the funtion, f (x) = x, the index is even, nd so the rdind must e greter thn or equl to 0. This tells us the domin is x 0 nd we write this in intervl nottion s [0, ). Previously we used point plotting to grph the funtion, f (x) = x. We hose xvlues, sustituted them in nd then reted hrt. Notie we hose points tht re perfet squres in order to mke tking the squre root esier.
88 80 Chpter 8 Roots nd Rdils One we see the grph, we n find the rnge of the funtion. The yvlues of the funtion re greter thn or equl to zero. The rnge then is [0, ). EXAMPLE 8.7 For the funtion f (x) = x +, find the domin grph the funtion use the grph to determine the rnge. Sine the rdil hs index, we know the rdind must e greter thn or equl to zero. If x + 0, then x. This tells us the domin is ll vlues x nd written in intervl nottion s [, ). To grph the funtion, we hoose points in the intervl [, ) tht will lso give us rdind whih will e esy to tke the squre root. Looking t the grph, we see the yvlues of the funtion re greter thn or equl to zero. The rnge then is [0, ). TRY IT : : 8.7 For the funtion f (x) = x +, find the domin grph the funtion use the grph to determine the rnge. TRY IT : : 8.8 For the funtion f (x) = x, find the domin grph the funtion use the grph to determine the rnge. In our previous work grphing funtions, we grphed f (x) = x ut we did not grph the funtion f (x) = do this now in the next exmple. EXAMPLE 8.7 x. We will This OpenStx ook is ville for free t
89 Chpter 8 Roots nd Rdils 8 For the funtion f (x) = x, find the domin grph the funtion use the grph to determine the rnge. Sine the rdil hs index, we know the rdind n e ny rel numer. This tells us the domin is ll rel numers nd written in intervl nottion s (, ) To grph the funtion, we hoose points in the intervl (, ) tht will lso give us rdind whih will e esy to tke the ue root. Looking t the grph, we see the yvlues of the funtion re ll rel numers. The rnge then is (, ). TRY IT : : 8.9 For the funtion f (x) = x, find the domin grph the funtion use the grph to determine the rnge. TRY IT : : 8.0 For the funtion f (x) = x, find the domin grph the funtion use the grph to determine the rnge. MEDIA : : Aess these online resoures for dditionl instrution nd prtie with rdil funtions. Domin of Rdil Funtion ( Domin of Rdil Funtion ( Finding Domin of Rdil Funtion (
90 8 Chpter 8 Roots nd Rdils 8.7 EXERCISES Prtie Mkes Perfet Evlute Rdil Funtion In the following exerises, evlute eh funtion.. f (x) = x, find. f (x) = 6x, find f () f (0). f () f ( ).. g(x) = 6x +, find g() g(8).. g(x) = x +, find g(8) g(). 7. G(x) = x, find G() G(). 60. g(x) = 7x, find g() g( ).. F(x) = x, find F() F( ). 8. G(x) = x +, find G() G(). 6. h(x) = x, find h( ) h(6). 6. F(x) = 8 x, find F() F( ). 9. g(x) = x, find g(6) g( ). 6. h(x) = x +, find h( ) h(6). 6. For the funtion f (x) = x, find f (0) f (). 6. For the funtion f (x) = x, find f (0) f (). 6. For the funtion g(x) = x, find g() g( ). 66. For the funtion g(x) = 8 x, find g( 6) g(). Find the Domin of Rdil Funtion In the following exerises, find the domin of the funtion nd write the domin in intervl nottion. 67. f (x) = x 68. f (x) = x 69. g(x) = x 70. g(x) = 8 x 7. h(x) = x 7. h(x) = 6 x + 7. f (x) = x + x 76. g(x) = 6x + 7. f (x) = x x f (x) = x 6 7. g(x) = 8x 78. f (x) = 6x This OpenStx ook is ville for free t
91 Chpter 8 Roots nd Rdils F(x) = 8x F(x) = 0 7x 8. G(x) = x 8. G(x) = 6x Grph Rdil Funtions In the following exerises, find the domin of the funtion grph the funtion use the grph to determine the rnge. 8. f (x) = x + 8. f (x) = x 8. g(x) = x g(x) = x 87. f (x) = x f (x) = x 89. g(x) = x 90. g(x) = x 9. f (x) = x 9. f (x) = x 9. g(x) = x 9. g(x) = x + 9. f (x) = x f (x) = x 97. g(x) = x g(x) = x 99. f (x) = x f (x) = x 0. g(x) = x 0. g(x) = x 0. f (x) = x 0. f (x) = x Writing Exerises 0. Explin how to find the domin of fourth root funtion. 06. Explin how to find the domin of fifth root funtion. 07. Explin why y = x Self Chek is funtion. 08. Explin why the proess of finding the domin of rdil funtion with n even index is different from the proess when the index is odd. After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. Wht does this heklist tell you out your mstery of this setion? Wht steps will you tke to improve?
92 8 Chpter 8 Roots nd Rdils 8.8 Use the Complex Numer System Lerning Ojetives By the end of this setion, you will e le to: Evlute the squre root of negtive numer Add nd sutrt omplex numers Multiply omplex numers Divide omplex numers Simplify powers of i Be Prepred! Before you get strted, tke this rediness quiz.. Given the numers, 7, 0., 7,, 8, list the rtionl numers, irrtionl numers, rel numers. If you missed this prolem, review Exmple... Multiply: (x )(x + ). If you missed this prolem, review Exmple.8.. Rtionlize the denomintor:. If you missed this prolem, review Exmple.. Evlute the Squre Root of Negtive Numer Whenever we hve sitution where we hve squre root of negtive numer we sy there is no rel numer tht equls tht squre root. For exmple, to simplify, we re looking for rel numer x so tht x =. Sine ll rel numers squred re positive numers, there is no rel numer tht equls when squred. Mthemtiins hve often expnded their numers systems s needed. They dded 0 to the ounting numers to get the whole numers. When they needed negtive lnes, they dded negtive numers to get the integers. When they needed the ide of prts of whole they dded frtions nd got the rtionl numers. Adding the irrtionl numers llowed numers like. All of these together gve us the rel numers nd so fr in your study of mthemtis, tht hs een suffiient. But now we will expnd the rel numers to inlude the squre roots of negtive numers. We strt y defining the imginry unit i s the numer whose squre is. Imginry Unit The imginry unit i is the numer whose squre is. i = or i = We will use the imginry unit to simplify the squre roots of negtive numers. Squre Root of Negtive Numer If is positive rel numer, then = We will use this definition in the next exmple. Be reful tht it is ler tht the i is not under the rdil. Sometimes you will see this written s = i to emphsize the i is not under the rdil. But the = i is onsidered stndrd form. EXAMPLE 8.76 Write eh expression in terms of i nd simplify if possile: 7. i This OpenStx ook is ville for free t
93 Chpter 8 Roots nd Rdils 8 Use the definition of he squre root of negtive numers. Simplify. Use the definition of he squre root of negtive numers. Simplify. Use the definition of he squre root of negtive numers. Simplify. i i 7 7i Be reful tht it is ler tht i is not under the rdil sign. i i TRY IT : : 8. Write eh expression in terms of i nd simplify if possile: 8 8. TRY IT : : 8. Write eh expression in terms of i nd simplify if possile: 6 7. Now tht we re fmilir with the imginry numer i, we n expnd the rel numers to inlude imginry numers. The omplex numer system inludes the rel numers nd the imginry numers. A omplex numer is of the form + i, where, re rel numers. We ll the rel prt nd the imginry prt. Complex Numer A omplex numer is of the form + i, where nd re rel numers. A omplex numer is in stndrd form when written s + i, where nd re rel numers. If = 0, then + i eomes + 0 i =, nd is rel numer. If 0, then + i is n imginry numer. If = 0, then + i eomes 0 + i = i, nd is lled pure imginry numer. We summrize this here.
94 86 Chpter 8 Roots nd Rdils + i = i Rel numer 0 + i Imginry numer = i i Pure imginry numer The stndrd form of omplex numer is + i, so this explins why the preferred form is = i when > 0. The digrm helps us visulize the omplex numer system. It is mde up of oth the rel numers nd the imginry numers. Add or Sutrt Complex Numers We re now redy to perform the opertions of ddition, sutrtion, multiplition nd division on the omplex numers just s we did with the rel numers. Adding nd sutrting omplex numers is muh like dding or sutrting like terms. We dd or sutrt the rel prts nd then dd or sutrt the imginry prts. Our finl result should e in stndrd form. EXAMPLE 8.77 Add: + 7. Use the definition of he squre root of negtive numers. Simplify the squre roots. Add. + 7 i + 7 i i + i i TRY IT : : 8. Add: 8 +. TRY IT : : 8. Add: Rememer to dd oth the rel prts nd the imginry prts in this next exmple. EXAMPLE 8.78 Simplify: ( i) + ( + 6i) ( i) ( i). This OpenStx ook is ville for free t
95 Chpter 8 Roots nd Rdils 87 Use the Assoitive Property to put the rel prts nd the imginry prts together. Simplify. Distriute. Use the Assoitive Property to put the rel prts nd the imginry prts together. Simplify. ( i) + ( + 6i) ( + ) + ( i + 6i) 9 + i ( i) ( i) i + i i + i i TRY IT : : 8. Simplify: ( + 7i) + ( i) (8 i) ( i). TRY IT : : 8.6 Simplify: ( i) + ( i) ( + i) ( 6i). Multiply Complex Numers Multiplying omplex numers is lso muh like multiplying expressions with oeffiients nd vriles. There is only one speil se we need to onsider. We will look t tht fter we prtie in the next two exmples. EXAMPLE 8.79 Multiply: i(7 i). i(7 i) Distriute. i 0i Simplify i. i 0( ) Multiply. i + 0 Write in stndrd form. 0 + i TRY IT : : 8.7 Multiply: i( i). TRY IT : : 8.8 Multiply: i( + i). In the next exmple, we multiply the inomils using the Distriutive Property or FOIL. EXAMPLE 8.80 Multiply: ( + i)( i). ( + i)( i) Use FOIL. 9i + 8i 6i Simplify i nd omine like terms. i 6( ) Multiply. i + 6 Comine the rel prts. 8 i TRY IT : : 8.9 Multiply: ( i)( i).
96 88 Chpter 8 Roots nd Rdils TRY IT : : 8.60 Multiply: ( i)( + i). In the next exmple, we ould use FOIL or the Produt of Binomil Squres Pttern. EXAMPLE 8.8 Multiply: ( + i) Use the Produt of Binomil Squres Pttern, ( + ) = + +. Simplify. Simplify i. Simplify. TRY IT : : 8.6 Multiply using the Binomil Squres pttern: ( i). TRY IT : : 8.6 Multiply using the Binomil Squres pttern: ( + i). Sine the squre root of negtive numer is not rel numer, we nnot use the Produt Property for Rdils. In order to multiply squre roots of negtive numers we should first write them s omplex numers, using = i. This is one ple students tend to mke errors, so e reful when you see multiplying with negtive squre root. EXAMPLE 8.8 Multiply: 6. To multiply squre roots of negtive numers, we first write them s omplex numers. 6 Write s omplex numers using = i. 6 i i Simplify. 6i i Multiply. i Simplify i nd multiply. TRY IT : : 8.6 Multiply: 9. TRY IT : : 8.6 Multiply: 6 8. In the next exmple, eh inomil hs squre root of negtive numer. Before multiplying, eh squre root of negtive numer must e written s omplex numer. EXAMPLE 8.8 This OpenStx ook is ville for free t
97 Chpter 8 Roots nd Rdils 89 Multiply: + 7. To multiply squre roots of negtive numers, we first write them s omplex numers. Write s omplex numers using = i. + 7 i + i Use FOIL. + 9 i 0 i 6 i Comine like terms nd simplify i. i 6 ( ) Multiply nd omine like terms. i TRY IT : : 8.6 Multiply: ( ) 8. TRY IT : : 8.66 Multiply: We first looked t onjugte pirs when we studied polynomils. We sid tht pir of inomils tht eh hve the sme first term nd the sme lst term, ut one is sum nd one is differene is lled onjugte pir nd is of the form ( ), ( + ). A omplex onjugte pir is very similr. For omplex numer of the form + i, its onjugte is i. Notie they hve the sme first term nd the sme lst term, ut one is sum nd one is differene. Complex Conjugte Pir A omplex onjugte pir is of the form + i, i. We will multiply omplex onjugte pir in the next exmple. EXAMPLE 8.8 Multiply: ( i)( + i). ( i)( + i) Use FOIL i 6i i Comine like terms nd simplify i. 9 ( ) Multiply nd omine like terms. TRY IT : : 8.67 Multiply: ( i) ( + i). TRY IT : : 8.68 Multiply: ( + i) ( i). From our study of polynomils, we know the produt of onjugtes is lwys of the form ( )( + ) =. The result is lled differene of squres. We n multiply omplex onjugte pir using this pttern. The lst exmple we used FOIL. Now we will use the Produt of Conjugtes Pttern.
98 80 Chpter 8 Roots nd Rdils Notie this is the sme result we found in Exmple 8.8. When we multiply omplex onjugtes, the produt of the lst terms will lwys hve n i whih simplifies to. ( i)( + i) (i) i ( ) + This leds us to the Produt of Complex Conjugtes Pttern: ( i)( + i) = + Produt of Complex Conjugtes If nd re rel numers, then ( i)( + i) = + EXAMPLE 8.8 Multiply using the Produt of Complex Conjugtes Pttern: (8 i)(8 + i). Use the Produt of Complex Conjugtes Pttern, ( i)( + i) = +. Simplify the squres. Add. TRY IT : : 8.69 Multiply using the Produt of Complex Conjugtes Pttern: ( 0i)( + 0i). TRY IT : : 8.70 Multiply using the Produt of Complex Conjugtes Pttern: ( + i)( i). Divide Complex Numers Dividing omplex numers is muh like rtionlizing denomintor. We wnt our result to e in stndrd form with no imginry numers in the denomintor. EXAMPLE 8.86 HOW TO DIVIDE COMPLEX NUMBERS Divide: + i i. This OpenStx ook is ville for free t
99 Chpter 8 Roots nd Rdils 8 TRY IT : : 8.7 Divide: + i i. TRY IT : : 8.7 Divide: + 6i 6 i. We summrize the steps here. HOW TO : : HOW TO DIVIDE COMPLEX NUMBERS. Step. Step. Step. Write oth the numertor nd denomintor in stndrd form. Multiply the numertor nd denomintor y the omplex onjugte of the denomintor. Simplify nd write the result in stndrd form. EXAMPLE 8.87 Divide, writing the nswer in stndrd form: + i. Multiply the numertor nd denomintor y the omplex onjugte of the denomintor. Multiply in the numertor nd use the Produt of Complex Conjugtes Pttern in the denomintor. Simplify. + i ( i) ( + i)( i) + 6i + + 6i 9 Write in stndrd form i
100 8 Chpter 8 Roots nd Rdils TRY IT : : 8.7 Divide, writing the nswer in stndrd form: i. TRY IT : : 8.7 Divide, writing the nswer in stndrd form: + i. Be reful s you find the onjugte of the denomintor. EXAMPLE 8.88 Divide: + i. i Write the denomintor in stndrd form. Multiply the numertor nd denomintor y the omplex onjugte of the denomintor. Simplify. + i i + i 0 + i ( + i)(0 i) (0 + i)(0 i) ( + i)( i) (i)( i) Multiply. 0i i 6i Simplify the i. 0i + 6 Rewrite in stndrd form i Simplify the frtions. i TRY IT : : 8.7 Divide: + i. i TRY IT : : 8.76 Divide: + i. i Simplify Powers of i The powers of i mke n interesting pttern tht will help us simplify higher powers of i. Let s evlute the powers of i to see the pttern. i i i i i i i i i i ( )( ) i We summrize this now. i i 6 i 7 i 8 i i i i i i i i i i i i i i i This OpenStx ook is ville for free t
101 Chpter 8 Roots nd Rdils 8 i = i i = i i = i 6 = i = i i 7 = i i = i 8 = If we ontinued, the pttern would keep repeting in loks of four. We n use this pttern to help us simplify powers of i. Sine i =, we rewrite eh power, i n, s produt using i to power nd nother power of i. We rewrite it in the form i n = i q i r, where the exponent, q, is the quotient of n divided y nd the exponent, r, is the reminder from this division. For exmple, to simplify i 7, we divide 7 y nd we get with reminder of. In other words, 7 = +. So we write i 7 = i nd then simplify from there. EXAMPLE 8.89 Simplify: i 86. Divide 86 y nd rewrite i 86 in the i n = i q i r form. i 86 i Simplify. () ( ) Simplify. TRY IT : : 8.77 Simplify: i 7. TRY IT : : 8.78 Simplify: i 9. MEDIA : : Aess these online resoures for dditionl instrution nd prtie with the omplex numer system. Expressing Squre Roots of Negtive Numers with i ( Sutrt nd Multiply Complex Numers ( Dividing Complex Numers ( Rewriting Powers of i (
102 8 Chpter 8 Roots nd Rdils Prtie Mkes Perfet Evlute the Squre Root of Negtive Numer In the following exerises, write eh expression in terms of i nd simplify if possile EXERCISES Add or Sutrt Complex Numers In the following exerises, dd or sutrt ( + i) + (7 + i) 8. (6 + i) + ( i) 9. (8 i) + (6 + i) 0. (7 i) + ( 6i). ( i) ( 6i). (8 i) ( + 7i). (6 + i) ( i). ( + i) ( + 6i) Multiply Complex Numers In the following exerises, multiply. 9. i( i) 0. i( + i). 6i( i). i(6 + i). ( + i)( + 6i). ( i)( + i). ( + i)( 7i) 6. ( 6 i)( i) In the following exerises, multiply using the Produt of Binomil Squres Pttern. 7. ( + i) 8. ( + i) 9. ( i) 0. ( 6 i) In the following exerises, multiply This OpenStx ook is ville for free t
103 Chpter 8 Roots nd Rdils ( 7) ( i)( + i) 0. ( i)( + i). (7 i)(7 + i). ( 8i)( + 8i) In the following exerises, multiply using the Produt of Complex Conjugtes Pttern.. (7 i)(7 + i). (6 i)(6 + i). (9 i)(9 + i) 6. ( i)( + i) Divide Complex Numers In the following exerises, divide i i 8. i + i 9. + i i 60. i 6 + i 6. i i 7i 6. i 6. + i 67. i i 6. i 6. + i i 68. i i Simplify Powers of i In the following exerises, simplify. 69. i 70. i 9 7. i i 8 7. i 8 7. i 6 7. i i Writing Exerises 77. Explin the reltionship etween rel numers nd omplex numers. 78. Aniket multiplied s follows nd he got the wrong nswer. Wht is wrong with his resoning? Why is 6 = 8i ut 6 =. 80. Explin how dividing omplex numers is similr to rtionlizing denomintor.
104 86 Chpter 8 Roots nd Rdils Self Chek After ompleting the exerises, use this heklist to evlute your mstery of the ojetives of this setion. On sle of 0, you improve this? how would you rte your mstery of this setion in light of your responses on the heklist? How n This OpenStx ook is ville for free t
Project 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationHS PreAlgebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem
HS PreAlger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.: Evlute squre roots
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationAlgebraic fractions. This unit will help you to work with algebraic fractions and solve equations. rs r s 2. x x.
Get strted 25 Algeri frtions This unit will help you to work with lgeri frtions nd solve equtions. AO1 Flueny hek 1 Ftorise 2 2 5 2 25 2 6 5 d 2 2 6 2 Simplify 2 6 3 rs r s 2 d 8 2 y 3 6 y 2 3 Write s
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 09765840 Volume 6, Numer (04), pp. 89 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis  The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationIntermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for rightngled tringles
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationPythagoras theorem and surds
HPTER Mesurement nd Geometry Pythgors theorem nd surds In IEEM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of rightngled tringle. This result is known s Pythgors
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1  Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A ... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationPythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.
Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationCS 573 Automata Theory and Formal Languages
Nondeterminism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Nondeterministi Finite Automton with moves (NFA) An NFA is tuple
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationAlgebra Basics. Algebra Basics. Curriculum Ready ACMNA: 133, 175, 176, 177, 179.
Curriulum Redy ACMNA: 33 75 76 77 79 www.mthletis.om Fill in the spes with nything you lredy know out Alger Creer Opportunities: Arhitets eletriins plumers et. use it to do importnt lultions. Give this
More informationQUADRATIC EQUATION EXERCISE  01 CHECK YOUR GRASP
QUADRATIC EQUATION EXERCISE  0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More information21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae
MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9 # + 4 ' ) ' ( 9 + 7) ' ' Give this
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationComputing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More informationEngr354: Digital Logic Circuits
Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimumost
More informationChapter 4 StateSpace Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 StteSpe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in nonright ngled tringles. This unit
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationBEGINNING ALGEBRA (ALGEBRA I)
/0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationMathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers
Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule
More informationUNCORRECTED SAMPLE PAGES. surds NUMBER AND ALGEBRA
Chpter Wht you will lern A Irrtionl numers n surs (0A) B Aing n sutrting surs (0A) C Multiplying n iviing surs (0A) D Binomil prouts (0A) E Rtionlising the enomintor (0A) F Review of inex lws (Consoliting)
More informationSection 3.1: Exponent Properties
Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted
More informationNumber systems: the Real Number System
Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationChapter33. Real numbers
Chpter Rel numers Contents: A Frtions B Opertions with frtions C Deiml numers D Opertions with deiml numers E Rtionl numers F Irrtionl numers 46 REAL NUMBERS (Chpter ) Opening prolem QiZheng hs irulr
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationK 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:
Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to
More informationExponentials  Grade 10 [CAPS] *
OpenStxCNX module: m859 Exponentils  Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More informationAlgebra 2 Semester 1 Practice Final
Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers
More informationFor a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs7.2ADV.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
More informationVectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a
Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly.
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. 6 ì 4 = 6 ì 0 = 4 ì 0 = 6 ì 3 = 5 ì 3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = 5 ì 2 = 6 ì 2 = 6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity B Integer multipliction doc69 Activity B More integer multipliction doc698 Activity B Advnced integer multipliction doc699 Multiplying integers FLUENCY
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationSection 3.2: Negative Exponents
Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationLesson 2.4 Exercises, pages
Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0  ( ) () 0 c) 7 + d) (7) ( ) 7  + 8 () ( 8). Expnd nd simplify. ) b)  7  + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationEquivalent fractions have the same value but they have different denominators. This means they have been divided into a different number of parts.
Frtions equivlent frtions Equivlent frtions hve the sme vlue ut they hve ifferent enomintors. This mens they hve een ivie into ifferent numer of prts. Use the wll to fin the equivlent frtions: Wht frtions
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PEhing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationGM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationTrigonometry and Constructive Geometry
Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties
More informationAT100  Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a
Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedystte nlysis of high voltge trnsmission systems, we mke use of the perphse equivlent
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationProbability. b a b. a b 32.
Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information