# Exponents

An exponent tells how many times to multiply a number (the ‘base’ number).

`3 ^{4} = (3)(3)(3)(3) = 81`

`x`

^{n}= (x)(x)(x)…(x is mulitpled *n* times)

Exponents are shorthand: it is much quicker to write 5^{4} than (5)(5)(5)(5)

### Examples:

`
3 ^{2} = (3)(3) = 9
3^{3} = (3)(3)(3) = 27
3^{4} = (3)(3)(3)(3) = 81
`

### Example: Solve 5^{4}

- Multiply the base by itself: (5)(5) = 25
- Multiply the product times the base (third time): (25)(5) = 125
- Multiply the product times the base (fourth time): (125)(5) =625

` 5 ^{2} = (5)(5) = 25
5^{3} = (5)(5)(5) = (25)(5) = 125
5^{4} = (5)(5)(5)(5) = (125)(5) = 625
5^{5} = (5)(5)(5)(5)(5) = (625)(5) = 3125
`

### Pattern of Positive and Negative Exponents

A positive base raised to a positive number gets larger as the exponent gets higher.

An exponential term with a negative exponent gets smaller as the exponent gets further below zero.

4^{4} |
(4·4·4·4) | 256 | ||

4^{3} |
(4·4·4) | 64 | ||

4^{2} |
(4·4) | 16 | ||

4^{1} |
4 | 4 | ||

4^{0} |
1 | 1 | ||

4^{-1} |
14^{1} |
14 | 14 | |

4^{-2} |
14^{2} |
1(4·4) | 116 | |

4^{-3} |
14^{3} |
1(4·4·4) | 164 | |

4^{-4} |
14^{4} |
1(4·4·4·4) | 1256 | |

1x |

### How are exponents useful?

Exponents are shorthand: it is much quicker to write 5^{4} than (5)(5)(5)(5).

Exponents can represent change at a constant rate, for example:

- A population doubling every year (e.g. population change rate = (p)(2
^{x}) where x = number of years)

- Base
- A base is a number or term that is raised to a power, or multiplied by itself. In 6x
^{3}, x is the base. In (6x)^{3}, 6x is the base. - Exponent
- An exponent tells how many times to multiply a number or term (the "base" ). The exponent is called a power. In 6
^{3}, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6). - Power
- An exponent. A power tells how many times to multiply a number or term. In 6
^{3}, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6). - Product
- The number or expression that is the result of multiplying numbers and/or expressions.
- Reciprocal
- The reciprocal of a number is 1 divided by that number. The reciprocal of x is 1x. This can also be written as x
^{-1}When you multiply a number by its reciprocal, the result is 1. e.g. (x)(1x) = 1. - Term
- A term is a number, variable, or the product of a number and variable(s). These are each terms: 4, x, 4x, 4x
^{2}. In an expression, terms are separated by addition or subtraction operators (+,-).

## Vocabulary

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.