Download Chapter 9: Powers and Roots

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.
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- - --
--
----
,-...
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