14.2 Simplifying Expressions with Rational Exponents and Radicals

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1 Name Class Date 14. Simplifyig Expressios with Ratioal Expoets ad Radicals Essetial Questio: How ca you write a radical expressio as a expressio with a ratioal expoet? Resource Locker Explore Explorig Operatios with Ratioal ad Irratioal Numbers What happes whe you add two ratioal umbers? Is the result always aother ratioal umber or ca it be irratioal? Will the sum of two irratioal umbers always be ratioal, always be irratioal, or ca it be either? What about the product of two irratioal umbers? These questios are all used to determie whether a set of umbers is closed uder a operatio. If the sum of two ratioal umbers is always ratioal, the set of ratioal umbers would be said to be closed uder additio. The followig tables will combie ratioal ad irratioal umbers i various ways. The various sums ad products should provide a geeral idea of which sets are closed uder the differet operatios. A Defie ratioal ad irratioal umbers. B Complete the followig additio table. Note that there are both ratioal ad irratioal addeds. + -π π -π - _ 3 -π C Based o the results i the table, will the sum of two ratioal umbers sometimes, always, or ever be a ratioal umber? Module Lesso

2 D What about the sum of two irratioal umbers? E Ad fially, the sum of a ratioal umber ad a irratioal umber? F Now complete the followig multiplicatio table. Similarly, it has both ratioal ad irratioal factors. -π π π _ 3 -π _ 3 _ 3 G H I Based o the results i the table, will the product of two ratioal umbers sometimes, always, or ever be a ratioal umber? What about the product of two irratioal umbers? Ad fially, the product of a ratioal umber ad a irratioal umber? Module Lesso

3 Reflect 1. Prove that the product of two ratioal umbers is a ratioal umber by cofirmig the geeral case.. Discussio Cosider the followig statemet: The product of two ratioal umbers is a irratioal umber. Is it a true statemet? Justify your aswer. Explai 1 Simplifyig Multivariable Expressios Cotaiig Radicals As you have see, to simplify expressios cotaiig radicals, you ca rewrite the expressios as powers with ratioal expoets. You ca use properties of expoets. You have already see the Power of a Power Property of expoets. There are additioal properties of expoets that are suggested by the followig examples. 3 = ( )( ) = 5 = +3 _ 3 = _ = 1 = 3- ( 3) = ( 3)( 3) = ( )(3 3) = 3 ( _ = _ 3 _ 3 = _ 3 3 = _ 3 3) ( 3 ) = ( ) = ( )( ) = 6 = 3 These relatioships are formalized i the table o the followig page. Module Lesso

4 Previous lessos have covered the properties of iteger expoets. A atural extesio of this is to ask if a umber ca be raised to a expoet that is a ratioal umber. The aswer is yes. If we defie a = a where is a iteger ad 0, we ca demostrate that a m_ =( a ) m whe m ad are itegers ad 0. a m = a 1 m.m = (a ) = ( a ) m Notice that a is ot defied if is eve ad a < 0. Example 1 A Properties of Expoets Let a ad b be real umbers ad m ad be ratioal umbers. Product of Powers Propertya m a = a m+ Quotiet of Powers Property a m _ a = a m-, a 0 Power of a Product Property (a b) = a b Power of a Quotiet Property ( a_ 9 3 (xy) 9 3 (xy) a b) = _ Power of a Power Property (a m ) b, b 0 = a m Simplify each expressio. Assume all variables are positive. 9_ = (xy) 3 Rewrite usig ratioal expoet. 3 = (xy) Simplify the fractio i the expoet. = x 3 y 3 Power of a Product Property B 5 x x 5 x _ x = x x Rewrite usig ratioal expoets. = x Product of Powers Property Reflect = x Simplify the expoet. = x Rewrite the expressio i radical form. 3. Discussio Why is a ot defied whe is eve ad a < 0? 4. Rewrite the expressio - a so that has a coefficiet of 1. The state the coditios uder which the expressio is udefied. Module Lesso

5 Your Tur Simplify each expressio. Assume all variables are positive. 5. (x y) 4 4 x y 4 6. _ 8 4 x 6 Explai Simplifyig Multivariable Expressios Cotaiig Ratioal Expoets Use Properties of Ratioal Expoets to simplify expressios. Example Simplify each expressio. Assume all variables are positive. ( A 8 x 9 _ ) 3 ( 8 x 9 ) 3 3 = ( ) 3 ( x 9 _ ) 3 Power of a Product Property = ( 3 _ 3) x (9 _ 3) Power of a Power Property = x 6 Simplify withi the paretheses. = 4 x 6 Simplify. B (64 x 1 ) 6 (64 x 1 ) 6 = ( ) ( x 1 ) Power of a Product Property Reflect = ( ) x ( ) Power of a Power Property = x Simplify withi the paretheses. = x Simplify. 7. Simplify ( 8 x 9 ) - _ 3. How is it related to the simplified form of ( 8 x 9 ) _ 3 foud i example A? Verify the relatioship if oe exists. Module Lesso

6 Your Tur Simplify each expressio. Assume all variables are positive. 8. ( 4 x x 1 ) - 9. ( 4 9 x x 4 ) 1 Explai 3 Simplifyig Real-World Expressios with Ratioal Expoets The relatioship betwee some real-world quatities ca be more complicated tha a liear or quadratic model ca accurately represet. Sometimes, i the most accurate model, the depedet variable is a fuctio of the idepedet variable raised to a ratioal expoet. Use the properties of ratioal expoets to solve the followig real-world scearios. Example 3 Biology Applicatio The approximate umber of Calories C that a aimal eeds each day is give by C = 7 m 3 4, where m is the aimal s mass i kilograms. A Fid the umber of Calories that a 65 kg bear eeds each day. To solve this, evaluate the equatio whe m = 65. C = 7 m = 7 (65) Substitute 65 for m. = 7 ( 4 65 ) 3 = 7 ( ) = 7 (5) 3 = 7 (15) = 9000 Defiitio of b m A 65 kilogram bear eeds 9000 Calories each day. Module Lesso

7 B A particular pada cosumes 1944 Calories each day. How much does this pada weigh? Substitute for C i the origial equatio ad solve for m. C = 7m 3 4 Origial equatio = 7m 3 4 Substitute for C. _ 4 = m 3 Divide each side by 7. 7 = m 3 4 Simplify. 3 3 = m 3 4 Rewrite the left side as a power. (3 3 ) = (m 3 4 ) Raise both sides to the power. 3 (3 ) = m 3 ( 4 ) Power of a Power Property 3 4 = m Simplify iside the paretheses. m = The pada weighs Simplify. kilograms. Your Tur Solve each real-world sceario. Image Credits: (t) DLILLC/ Corbis; (b) Radius Images/Corbis 10. The speed of light is the product of its frequecy f ad its wavelegth w. I air, the speed of light is m/s. a. Write a equatio for the relatioship described above, ad the solve this equatio for frequecy. b. Rewrite this equatio with w raised to a egative expoet. c. What is the frequecy of violet light whe its wavelegth is approximately 400 aometers (1 m = 10-9 m)? Module Lesso

8 11. Geometry The formula for the surface area of a sphere S i terms of its volume V is S = (4π) _ 3 (3V) 3. What is the surface area of a sphere that has a volume of 36π cm cubed? Leave the aswer i terms of π. What do you otice? Elaborate 1. A set of elemets is said to be closed uder some operatio if performace of that operatio o elemets of the set always produces a elemet of the set. Examie the set of itegers ad the set of ratioal umbers. Is each set closed uder each of the followig operatios: additio, multiplicatio, divisio, ad subtractio? Provide a couterexample if the set is ot closed uder a operatio. 13. Why are itegers closed uder multiplicatio? 14. Is the set of all umbers of the form a x, where a is a positive costat ad x is a ratioal umber, closed uder multiplicatio? Justify your aswer. 15. Essetial Questio Check-I How ca you write a radical expressio as a power with a ratioal expoet? Module Lesso

9 Evaluate: Homework ad Practice 1. Why are the additio ad multiplicatio tables i the Explore activity symmetric about the diagoal from the upper-left corer to the lower-right? For example, why is the etry i the third row of the secod colum equal to the etry i the secod row of the third colum? Would a subtractio table be symmetric about the same diagoal? Olie Homework Hits ad Help Extra Practice. Prove that the ratioal umbers are closed uder additio. Simplify the give expressio (7 x 3 ) (8 x 3 ) 5. 3 (8 y 3 ) 4 6 (8 y 3 ) x 7. ( x ) _ y 8. _ 8x _ 3 16x Module Lesso

10 _ 3 9. (0x) , z + 10, 000 z 11. ( 1 5x x 9 ) - 1. ( 15x 3 ) - _ (x) x (x) x x 14. (1,000,000x 6 ) (x y) 3 (x y y) 4 _ _ (x _ y) (x _ 8 y _ 4 z _ ) ( _ z y x ) 8 (x 10 ) _ 5 (x 19. _ ) _ x x 8 6 x 4 Module Lesso

11 1. (x y) 4 ( _ y _ ). ( y 8 4 ) 3. Biology Biologists use a formula to estimate the mass of a mammal s brai. For a mammal with a mass of m grams, the approximate mass B of the brai, also i grams, is give by B = _ m 3 8. Fid the approximate mass of the brai of a mouse that has a mass of 64 grams. 4. Multi-Step Scietists have foud that the life spa of a mammal livig i captivity is related to the mammal s mass. The life spa i years L ca be approximated by the formula L = 1m _ 5, where m is the mammal s mass i kilograms. How much loger is the life spa of a lio compared with that of a wolf? Image Credites: VisiosofAmerica.com/Joe Sohm Typical Mass of Mammals Mammal Mass (kg) Koala 8 Wolf 3 Lio 43 Giraffe 104 Module Lesso

12 Tim ad Tom are paiters. Use the give iformatio to provide the desired estimate. 5. Tim ad Tom use a liters of pait o a large shippig crate. If the ext crate they eed to pait is similar but has twice the volume, how much pait should they pla o buyig? 6. Tim ad Tom are paitig a crate. Tom paits 10 square feet per miute. They paited a particular crate i 1 day. Tim uses a sprayer ad is 4.7 times as fast as Tom. How log would it take them to pait a crate with twice the volume ad of similar shape? 7. Determie whether each of the followig are ratioal or irratioal. Select the correct aswer for each lettered part. a. The product of ad 50 Ratioal Irratioal b. The product of _ ad _ 5 Ratioal Irratioal c. C = πr evaluated for r = π -1 Ratioal Irratioal d. C = πr evaluated for r = 1 Ratioal Irratioal e. A = πr evaluated for r = π Ratioal Irratioal f. The product of _ π ad _ 50π Ratioal Irratioal g. The product of ad 9_ Ratioal Irratioal Module Lesso

13 H.O.T. Focus o Higher Order Thikig 8. Explai the Error Jim tried to show how to write a radical expressio as a power with a ratioal expoet. Suppose that - a = a k. (- a ) = (a k ) Raise each side to the th power. a = (a k ) Defiitio of th root a = a k Power of a Power Property a 1 = a k 1 = k Equate expoets. k = Solve for k. Jim claimed to have show that a = a. Explai ad correct his error. 9. What If? Assume the itegers are ot closed uder additio. a. Are the ratioal umbers closed uder multiplicatio? b. Are the ratioal umbers closed uder additio? Module Lesso

14 30. Commuicate Mathematical Ideas Prove by cotradictio that a ratioal umber plus a irratioal umber is irratioal. To do this assume the egatio of what you are tryig to prove ad show how it will logically lead to somethig cotradictig the give. Assume that a ratioal umber plus a irratioal umber is ratioal. r 1 + i 1 = r Give r 1 + i 1 - r 1 = r - r 1 Subtract r 1 from both sides. i 1 = r - r 1 Simplify left side. Provide the cotradictio statemet to fiish the proof. 31. Critical Thikig Show that a umber raised to the _ power is the same as the cube root of that umber. 3 Lesso Performace Task The balls used i soccer, baseball, basketball, ad golf are spheres. How much material is eeded to make each of the balls i the table? The formula for the surface area of a sphere is S A = 4πr ad the formula for the volume of a sphere is V = 4 3 πr 3. Use algebra to fid the formula for the surface area of a sphere give its volume. Complete the table with the surface area of each ball. Ball Volume (i cubic iches) Surface Area (i square iches) soccer ball baseball 1.8 basketball golf ball.48 Module Lesso