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Chapter 5—Powers, Exponents, and Roots

5-1 Powers and Exponents

Recall that multiplication provides a convenient way to express the need for adding the same number to itself over and over again.

Consider how 2 + 2 + 2 + 2  = 8
can be more conveniently expressed as 4 x 2 = 8.

Powers and exponents provide a convenient way for expressing a need for multiplying a particular number over and over again.

You will learn in this lesson that 4x 4 x 4 = 64
can be more conveniently expressed as 43.

Power Notation

Definition

Power notation—the method for indicating the power of a number—has two parts:
  • The base indicates the number to be multiplied.
  • The exponent indicates the number of times the base is to be multiplied.

fig0701_01.jpg (5380 bytes)

Terminology for powers and exponential notation.

Consider this example: 23

  • It is the same as 2 x 2 x 2
  • It is equal to 8
  • The base is 2 and the exponent is 3
  • It is spoken as "Two to the third power" or "Two to the power of three."

More Examples

  1. 32 = 3 x 3 = 9
  2. 24 = 2 x 2 x 2 x 2 = 16
  3. 45 = 4 x 4 x 4 x 4 x 4 = 1024

Note

  • A number with an exponent of 2 is often said to be squared.   A value such as 52 can be called "five squared."
  • A number with an exponent of 3 is often said to be cubed. A value such as 63 can described as "six cubed."

There are no such common expressions for numbers raised to any other power.

Examples & Exercises

Evaluating Powers

Use these interactive Examples & Exercises to strengthen your understanding and build your skills.

Special Cases

You must be aware of four special cases for expressions of powers and exponential notation.

  1. Any number with an exponent of 1 is equal to that number, itself.

Examples

51 = 5 81 = 8 1251 = 125
  1. Any number with an exponent of 0 is equal to 1.

    Examples

3 0 = 1 12 0 = 1 373 0 = 1
  1. 1 to any power is equal to 1.

    Examples

1 4 = 1 1 3 = 1 1 15 = 1
  1. 0 to any power is equal to 0.

    Examples

0 5 = 0 0 1 = 0 0 575 = 0

Summary

n1 = n
n0 = 1
1n = 1
0n = 0

Examples & Exercises

Special Cases of Exponents

Work these examples of special cases until you can do the work without making any errors. It is important that you NOT use a calculator for this exercise.

Evaluating Powers With Negative Exponents

Any number with a negative exponent is equal to 1 divided by that number with a positive exponent.

Examples
1.

2-3 =

1 = 1 = 0.125
23 8
2.

4-2 =

1 = 1 = 0.0625
42 16
3.

1-8 =

1 = 1 = 1   Note: 1 to any power is equal to 1.
18 1

Examples & Exercises

Negative Exponents

Evaluate these powers that have negative exponents. Express your answer as a fraction. It is important that you NOT use a calculator for this exercise.

Exponents That are Fractions or Decimal Values

You have learned about exponents that are positive or negative integers, exponents that are equal to 1, and exponents that are equal to 0. But what about exponents that are fractions or decimals? Here are some examples:

  • Three to the one-half power:  3
  • Twenty-two to the three-fourths power: 223/4
  • Six to the minus one-third power:  6 -1/3
  • Five to the 3.5 power:  53.5

Note

In order to understand and use exponents that are fractions or decimals, you must first know about roots of numbers. These are discussed in another lesson, and you will be reminded of fraction/decimal exponents at that time.

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015