Algebra 2: Section 7.1
Nth Roots and Rational
Exponents
1
7.1: Evaluating nth Roots
• You are familiar with square roots
– 2 is the square root of 4, since 22 = 4.
• This concept can be extended to other
types of roots
– 2 is the cube root of 8, since 23 = 8
– 2 is the fourth root of 16, since 24 = 16
– 2 is the fifth root of 32, since 25 = 32
– 2 is the nth root of a, since 2n = a
1
n
Rational Exponents
a  a
n
m
 
m
n

a

  ( a)
a
 
m
1
1

n  m  n
m
a
(
a
)
n
a
m
n
1
n
3
Examples
• Rewrite using rational exponents.
1.
2.
3
27  27
5
a a
1
3
3.
 7
4.
 3
5
1
5
8
2
7
5
2
2
8
3 3
• Rewrite using radical notation.
5. 7
4
3

 7
3
4
4
5
6. 4 
 4
5
4
4
1
4
Nth Roots
Rules for Signs of Answers
• For odd roots (3, 5, 7, etc)
– There is ALWAYS ONLY ONE answer
(either + or -)
• For even roots (2, 4, 6, etc)
– If the radical symbol is already in the
problem, then use the sign that is in front
– If YOU put the radical symbol in the
problem to solve an equation, then there
will be two answers (+ and -)
5
Examples
•
Find the nth root(s) of a
7.
n = 5, a = -32
-2
What multiplied by itself 5 times equals -32?
8.
n = 3: a = 64
4
What multiplied by itself 3 times equals 64?
9.
n = 4: a = 256
4
What multiplied by itself 4 times equals 256?
10. n = 2: a = 169
±13
What multiplied by itself 2 times equals 169?
6
Examples
Simplify or evaluate each expression:
3
11.
5
125
12.

13.
 0
4
6
16

2
3
 2
2
 0
3
 4
 0
7
Rational Exponents (examples)
•
Simplify or evaluate each expression:
14.
15.
16.
82/3 
163/4
4-5/2
 
3
2

 

1
4
3
16
 4
2
5
4
 64
3
8
 2
 8
1
1

32
3

 2 
5
Examples
• Solve the equation. Round your
answer to two decimal places.
3
17. 5x = 40
x 8
3
x2
9
Examples
18. ( y  1)  32
3
y  1  32
3
3
19. 3(x +5) = 81
 x  5
3
 27
y  32  1
x 5  3
 4.17
x  2
3
10
Homework
P.404
#13-22 all
#24-40 evens
#53-61 all
11
List of Perfects
Perfect Squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
144, 169, 196, 225
Perfect Cubes:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Perfect Quads:
1, 16, 81, 256, 625
Perfect Fifths:
1, 32, 243, 1024, 3125