RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals
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1 RADICALS m 1
2 RADICALS Upo completio, you should be ble to defie the pricipl root of umbers simplify rdicls perform dditio, subtrctio, multiplictio, d divisio of rdicls Mthemtics Divisio, IMSP, UPLB
3 Defiitio: th Root Suppose b. The we cll b th root of. Exmples: 8 - is the cube root of -8 4 d - re the squre roots of 4 Mthemtics Divisio, IMSP, UPLB
4 Defiitio: Suppose b. NOTICE THAT WE ARE NOT YET USING HERE THE RADICAL SIGN th Root The we cll b th root of. Exmples: 8 - is the cube root of -8 4 d - re the squre roots of 4 Mthemtics Divisio, IMSP, UPLB 4
5 th Root Notice tht if is odd, the we oly hve oe th root. But if is eve, the we hve two th roots (the positive th root d the egtive th root). Exmple: d 4 Mthemtics Divisio, IMSP, UPLB 5
6 Defiitio: Pricipl th Root Let be positive eve iteger d > 0, To remove the mbiguity, we defie to be the positive root. We cll this the pricipl th root. Hece, 4 d ot. Also, is clled the rdicd is the idex of the rdicl. Mthemtics Divisio, IMSP, UPLB 6
7 Pricipl th Root FYI: Let be positive eve or odd iteger 0 0. Mthemtics Divisio, IMSP, UPLB 7
8 Pricipl th Root I summry, the rdicl sig deotes the pricipl th root of umber. Mthemtics Divisio, IMSP, UPLB 8
9 A rdicl or (irrtiol expressio) is lgebric expressio ivolvig o-itegrl rtiol expoets. It is of the form Exmples: Write s rdicl. 1. 1/ /4 b m m or 4 b RADICALS. 1/ 1/ x y 1/ 1/ x y x x y y Mthemtics Divisio, IMSP, UPLB 9
10 RADICALS Exmples: Write the followig rdicls i expoetil form. 1. xy z 1/ / 1/ x y z. 4 x x 6 y y 5 1/ /4 x y 1/ 5/6 x y Mthemtics Divisio, IMSP, UPLB 10
11 Mthemtics Divisio, IMSP, UPLB 11 DEFINITION: If m d re NOT both eve. RADICALS m m m m m m 1 1 They re equl.
12 Mthemtics Divisio, IMSP, UPLB 1 Exmples: If m d re NOT both eve. RADICALS 1 1 odd eve
13 Mthemtics Divisio, IMSP, UPLB 1 DEFINITION: If m d re both eve. RADICALS m m m m m m 1 1 Notice the differece if m is iside or outside the prethesis.
14 Exmples: If m d re both eve. RADICALS eve eve Mthemtics Divisio, IMSP, UPLB 14
15 Exmples: If m d re both eve. RADICALS 4 x x 4 x 1 x Mthemtics Divisio, IMSP, UPLB 15
16 RADICALS DEFINITION: Negtive Rtiol Expoet m 1 m Mthemtics Divisio, IMSP, UPLB 16
17 Properties of Rdicls Theorem: If d b re rel umbers, the 1/ 1/ 1/ 1. b ( b) b b. 1/ 1/ b b b 1/ b, b 0 (where both > 0 d b > 0 if is eve). Mthemtics Divisio, IMSP, UPLB 17
18 Properties of Rdicls Notice the cluse: where both > 0 d b > 0 if is eve. Becuse: FYI: sqrt(-1) is imgiry umber Mthemtics Divisio, IMSP, UPLB 18
19 Properties of Rdicls Exmples: Use the properties to fid the ff Mthemtics Divisio, IMSP, UPLB 19
20 Properties of Rdicls Theorem: m m Mthemtics Divisio, IMSP, UPLB 0
21 Simplifyig Rdicls A rdicl is simplified if the followig hold: 1. There is o power i the rdicd higher th or equl to the idex. Exmples: Simplify x 4 x 4 x x x y z x y y z z 4 4 x y z y z Mthemtics Divisio, IMSP, UPLB 1
22 Simplifyig Rdicls A rdicl is simplified if the followig hold:. The idex d the expoets i the rdicd must hve o commo fctor. Exmples: Simplify r s t 4 s 6 r t 4 s r t s r t Mthemtics Divisio, IMSP, UPLB
23 Simplifyig Rdicls A rdicl is simplified if the followig hold:. There is o deomitor i the rdicd. Exmples: Simplify. 5. x y x y x y 15z 7 6 x y 15z 7 x z y 15z y Mthemtics Divisio, IMSP, UPLB
24 Multiplyig Rdicls RECALL: b b Exmples: Fid these products d simplify x 70x Assume vribles re greter th zero.. 4x y 6x z 45x y z Mthemtics Divisio, IMSP, UPLB 4
25 Multiplyig Rdicls How do you multiply rdicls with differet idices? Exmples: Fid these products d simplify Assume vribles re greter th zero. 4. x y z Mthemtics Divisio, IMSP, UPLB 5
26 Dividig Rdicls RECALL: b, b 0 b Exmples: Fid the quotiets d simplify x y xy Assume vribles re greter th zero.. 6 8x y 15z 7 Mthemtics Divisio, IMSP, UPLB 6
27 Dividig Rdicls Wht if the idices re ot the sme? Exmples: Fid the quotiets d simplify. 1. Assume vribles re greter th zero.. 4 x xy Mthemtics Divisio, IMSP, UPLB 7
28 To rtiolize the deomitor mes to get rid of rdicls i the deomitor. Exmples: Rtiolize the deomitor. 1. Rtiolizig the Deomitor Multiply umertor d deomitor by rtiolizig fctor Mthemtics Divisio, IMSP, UPLB 8
29 Rtiolizig the Deomitor Exmples: Rtiolize the deomitor.. 5x yz x y z Assume vribles re greter th zero. 4. b b Mthemtics Divisio, IMSP, UPLB 9
30 Rtiolizig the Deomitor Exmples: Rtiolize the deomitor x y 1 b bc c Assume vribles re greter th zero. Mthemtics Divisio, IMSP, UPLB 0
31 Similr Rdicls Rdicls re similr if they hve the sme idex d rdicds whe simplified. Exmples: Which of the followig pirs of rdicls re similr? Assume x> x, 1x 6. x, 4x 1., 8x x 5 Mthemtics Divisio, IMSP, UPLB 1
32 Addig d Subtrctig Rdicls We c oly dd(subtrct) similr rdicls. To do tht, dd(subtrct) their coefficiets d ffix the commo rdicl. Exmples: Fid the followig sums. Assume x> x 1x 6. x 4x 1. 8x x 5 x Mthemtics Divisio, IMSP, UPLB
33 Addig d Subtrctig Rdicls 1 4. x y x y x xy x y xy xy y x xy y xy 1 xy x y xy
34 Addig d Subtrctig Rdicls x xy y xy x y xy x y xy x y 1 xy xy 1 xy xy xy 1 x y xy xy xy
35 RECALL A rdicl is simplified if the followig hold: 1. There is o power i the rdicd higher th or equl to the idex.. The idex d the expoets i the rdicd must hve o commo fctor.. There is o deomitor i the rdicd. Mthemtics Divisio, IMSP, UPLB 5
36 COMMON MISTAKES 1. b b. b b Mthemtics Divisio, IMSP, UPLB 6
37 Exercises A. Fid the product d simplify the result: b b. 4x y 6x z 45x y z 4 5. b b Assume vribles re greter th zero. Mthemtics Divisio, IMSP, UPLB 7
38 Exercises B. Fid the quotiet d simplify the result: 1. 5x yz x y z Assume vribles re greter th zero.. 8 7ts 10r 5 Mthemtics Divisio, IMSP, UPLB 8
39 Exercises B. Fid the quotiet d simplify the result: Assume vribles re greter th zero.. 1x y Mthemtics Divisio, IMSP, UPLB 9
40 Exercises C. Rtiolize the deomitor d simplify the result: Mthemtics Divisio, IMSP, UPLB 40
41 Exercises D. Rtiolize the umertor d simplify the result: 1.. x 4 x x h x h Assume vribles re greter th zero. Mthemtics Divisio, IMSP, UPLB 41
42 Exercises E. Add or subtrct the followig rdicls d simplify: Mthemtics Divisio, IMSP, UPLB 4
43 Exercises E. Add or subtrct the followig rdicls d simplify: b 81 b 4. x y 1 y x xy 4 x y Assume vribles re greter th zero. Mthemtics Divisio, IMSP, UPLB 4
44 Reflectio 1. Wht is rdicl?. Wht opertios c be performed o rdicls? How do you do them?. Whe c you dd or subtrct rdicls? 4. Nme uses of rdicls i the rel world. Mthemtics Divisio, IMSP, UPLB 44
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