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Chapter 9: Powers and Roots 11. a. 272/3= (271/3i= 32= 9 Section 9.1 b. 163/4= (161/4)3 = 23= 8 1. 16-1 =~ 16 16° = 1 161=16 c. .00032 a. 1 < 161/2< 16 d. (J'2 25 b. Jl6 e. 129.64 c. -4 does not lie between 1 and 16, see part (a). d. No, the solutionsofx2= 16arex = 4 and x = -4, whereas the solution ofx =Jl6 1 1.J5 .J5 13. a. .J5 = .J5 *.J5 =5 is only x=4. b 2- _ 3fi _ 3fi . fi- fi*fi- 3. a. 75/2= (7112Y =(.J7r 75/2=(75Y'2=9 or d Jl4 .../2*7*.J5J70 . .J5= .J5*.J5 -5 b. ~ e. 5. ~C5i = 5 , but (-Nf f. is undefmed. The two expressions differ in the placement of the parentheses. In the first, we square -5 before taking the square root. In the second, we must attempt the square root oC5 before squaring. Negative numbers, however, do not have real number square roots. = -5 and ({[5Y =-5 Both these expressions result in taking a cube root of a negative number. Because the cube of a negative number is negative, the cube root of a negative number is a negative number. Section 9.2 provides further information on even and odd roots. 9. Each side of the cube is exactly ifiO cm or approximately 2.7 cm. "7 .J7 J3 =4 = J3 J7 *.J7 *.J7 -..J2i 7 j{=.J5=.J5*J6=J30 6 J6 J6 *J6 6 15. a. 75/2=(71/2Y =(71/2 *7112)*(71/2*i/2)*7112 = 7 * 7.J7 = 49.J7 b. 75/2 =49.J7 "" 49 * 2.6458 129.64 "" Skills and Review 9.1 17. a. 0 b. 3 is not in the domain of g(x); g(3) =.../3- 4 =-Fi , and negative numbers do not have real square roots. 21. b. 216 = 2*2*2*3*3*3; therefore 2161/3= 2*3 = 6 d. 32 = 2*2*2*2*2; therefore 321/5= 2 #= .o 19. 7. a. 16 = 2*2*2*2; therefore 161/4= 2. c. 64 = 2*2*2*2*2*2; therefore 641/3= 2*2 2 Jl2 .../2*2*3 2J3_? J3 -C. J3 = J3 or in simpleradicalform, 49.J7 VC5)) = 253/2 9)/2 = (251/2 (91/2tt =f.=E53 125 c2 +8c = -15 c2 +8c + 15 = 0 (c + 5)(c + 3) = 0 c+5=Oorc+3=0 .. Factor the left side Apply the Zero Product Property c = -5 or c = -3 An alternative is to use the quadratic formula. @ Houghton Mifflin Company. All rights reserved. - - -- -- ---- ,-... III! ------- 23. a. J? - lOx + 24 Factor pair of the diagonal product, 24J?,that sums to -lOx is -4x and -6x. = x2 - 4x - 6x + 24 Split the middle term = (J?- 4x) + C6x + 24) Form groups of two terms = x(x - 4) - 6(x - 4) Factor the GCF from each group =(x - 4)(x - 6) Factor out the common binomial, (x - 4) b. -4J?+ 36 = -4(x2- 9) = -4(x - 3)(x + 3) c. IOJ?- 35x = 5x(2x - 7) Factor the GCF from both terms Use the difference of two squares Factor the GCF from both terms 25. 6x -7(1 + x)=5 - 3x 6x -7 - 7x = 5 - 3x Use the distributive property. LS -x - 7 = 5 - 3x Combine like terms, LS 2x-7=5 Add 3x on both sides 2x = 12 Add 7 on both sides x=6 Divide by 2 on both sides