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

The number written outside of the radical sign is called the index and tells you what root you are looking for.

3 8

Rational Exponents

16Roots can also be written as rational

exponents. For example: 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

2

3

9

Rational exponents do not have to have a numerator of 1. For instance:

• 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

mn a

Let a 1/n be an nth root of a, and let m be a positive integer.

a m/n = (a 1/n)m =

a -m/n = mn

mnnmaaa

1

)(

11/1/

Some Examples

2

3

9

• Evaluate the following:

1.

2. 5

2

32

You Try

2

5

16

• Evaluate the following:

1.

2.

3

2

64

Using a Calculator

34 )5(

• Evaluate

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