Nth Root and Rational
Exponents
Section 7.1
WHAT YOU WILL LEARN:
1. How to evaluate nth roots of real numbers using
both radical notation and rational exponent
notation.
2. How to use nth roots to solve real-life problems.
Roots Other than Squares
23 is 8
We can also say the cube root of 8 is 2. (What
number times itself 3 times is 8)
This is more commonly written
3
8
The number written outside of the radical sign is
called the index and tells you what root you are
looking for.
Rational Exponents
Roots can also be written as rational
exponents. For example: 16 can be
rewritten as 16 ½
How do you think you would rewrite the
following? 3 64
Some Examples
• Find the indicated real nth root(s) of a.
1. n = 3, a = -125
2. n = 4, a = 16
More About Rational Exponents
Rational exponents do not have to have a
numerator of 1. For instance:
3
92
• The denominator is the index or root of the
number under the radical.
• The numerator is the power that you raise the
“answer” to.
“Mathy” Definition
Let a 1/n be an nth root of a, and let m be a
positive integer.
a
a
m/n
= (a
-m/n
=
1/n)m
1
a
m/ n
=

n
a
m
1
1/ n m
(a )
1

n
a
m
Some Examples
• Evaluate the following:
1.
2.
9
3
2
2

5
32
You Try
• Evaluate the following:
1. 16
2.
5
2
2

3
64
Using a Calculator
• Evaluate
( 4 5 )3
Solving Equations with Roots
• Solve the following:
1. 2x4 = 162
2. (x – 2)3 = 10
You Try
• Solve the following:
1. 6x4 = 3750
2. (x + 1)3 = 18
A Lovely Word Problem
• The volume for the volume V of a regular
dodecahedron is v = 7.66a3 where a is the length of an
edge of the dodecahedron. Find the length of an edge
of a regular dodecahedron that has a volume of 30
cubic feet.
Homework
Homework: page 404, 14-22
even, 30-44 even, 48, 54-60
even, 66