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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 4^{3}.

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. | Terminology for powers and exponential notation. | |

Consider this example: 2^{3}

- 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

- 3
^{2} = 3 x 3 = 9 - 2
^{4} = 2 x 2 x 2 x 2 = 16 - 4
^{5} = 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 5^{2} can be called "five squared." - A number with an exponent of 3 is often said to be
**cubed**. A value such as 6^{3} 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.

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

Examples

5^{1} = 5 | 8^{1} = 8 | 125^{1} = 125 |

**Any number with an exponent of 0 is equal to 1.**Examples

3^{ 0} = 1 | 12^{ 0} = 1 | 373^{ 0} = 1 |

**1 to any power is equal to 1.**Examples

1^{ 4} = 1 | 1^{ 3} = 1 | 1^{ 15} = 1 |

**0 to any power is equal to 0.**Examples

0^{ 5} = 0 | 0^{ 1} = 0 | 0^{ 575} = 0 |

- n
^{1} = n - n
^{0} = 1 - 1
^{n} = 1 - 0
^{n} = 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 |

2^{3} | 8 |

2. | 4^{-2} = | 1 | = | 1 | = 0.0625 |

4^{2} | 16 |

3. | 1^{-8} = | 1 | = | 1 | = 1 Note: 1 to any power is equal to 1. |

1^{8} | 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
^{½} - Twenty-two to the three-fourths power: 22
^{3/4} - Six to the minus one-third power: 6
^{-1/3} - Five to the 3.5 power: 5
^{3.5}

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. |