Download nth Roots and Rational Exponents

7.1
nth Roots and Rational
Exponents
Goals p Evaluate nth roots using radical notation and rational
exponent notation.
p Use nth roots to solve real-life problems.
Your Notes
VOCABULARY
nth root of a For an integer n greater than 1, if bn ! a,
then b is an nth root of a.
n
Index of a radical The integer n in the expression !a
",
where n > 1
REAL nth ROOTS
Let n be an integer greater than 1 and let a be a real
number.
n
• If n is odd, then a has one real nth root: !a
" ! a1/n
• If n is even and a > 0, then a has two real nth roots:
n
"!a
" ! "a1/n
• If n is even and a ! 0, then a has one nth root:
n
!0
" ! 01/n ! 0
• If n is even and a < 0, then a has no real nth roots .
152
Algebra 2 Notetaking Guide • Chapter 7
Your Notes
Finding nth Roots
Example 1
Find the indicated real nth root(s) of a.
a. n ! 3, a ! 64
b. n ! 4, a ! 81
Solution
a. Because n ! 3 is odd, a ! 64 has one real cube root .
Because 4 3 ! 64, you can write:
3
!64
"! 4
641/3 ! 4
or
b. Because n ! 4 is even and a ! 81 > 0, 81 has
two real fourth roots . Because 3 4 ! 81 and
(#3)
4
! 81, you can write:
4
" ! "3
"!81
or
"811/4 ! "3
Checkpoint Find the indicated nth root(s) of a.
1. n ! 3, a ! 343
2. n ! 4, a ! 256
7
"4
RATIONAL EXPONENTS
Let a1/n be an nth root of a, and let m be a positive integer.
n
• am/n ! (a1/n)m ! ( !a
" )m
1
1
• a#m/n ! 
!$
!
m/n
1/n
a
(a )m
Example 2
a. 253/2 !
1
n
"
( !a
,a%0
)m
Evaluating Expressions with Rational Exponents
"$3 ! 53 ! 125
#!25
1
1
1
1
$
2 !$ ! $
3
!
b. 27#2/3 ! $
2
2/3
3
9
27
27$
#!"
Lesson 7.1 • Algebra 2 Notetaking Guide
153
Your Notes
Solving Equations Using nth Roots
Example 3
a. 3x6 ! 192
b. (x & 3)5 ! 11
x6 ! 64
x&3!
6
5
!11
"
5
x ! "!64
"
x!
x ! "2
x % #1.38
Example 4
!11
"#3
Evaluating a Model with nth Roots
The population P of a certain animal species after t years can
be modeled by P ! C(2.72)#t/3 where C is the initial
population. Find the population after 2 years if the initial
population was 1000.
Solution
P ! C(2.72)#t/3
Write model for population.
! 1000(2.72)#2/3
Substitute for C and t.
% 513
Use a calculator.
The population of the species is about 513 after 2 years.
Checkpoint Complete the following exercises.
3. Evaluate 16#3/4.
1
$$
8
Homework
5
!34
" & 5 ≈ 7.02
0.1942x#1/3
5. Solve y ! $$
for y when x ! 0.3 and z ! 4.
x3z5/2
about 0.34
154
4. Solve (z # 5)5 ! 34.
Algebra 2 Notetaking Guide • Chapter 7