# Negative Exponents

Exponents are also called **Powers** or **Indices**

Let us first look at what an "exponent" is:

The In this example: |
||

In words: 8^{2} can be called "8 to the second power", "8 to the power 2"or simply "8 squared" |

### Example: **5**^{3} = 5 × 5 × 5 = 125

^{3}= 5 × 5 × 5 = 125

In words: 5^{3} can be called "5 to the third power", "5 to the power 3" or simply
"5 cubed"

**In general**:

a^{n} tells you to use a in a multiplication
n times: |

But those are **positive exponents**, what about something like:

8^{-2}

That exponent is **negative** ... what does it mean?

## Negative Exponents

Negative? What could be the opposite of multiplying? **Dividing!**

Dividing is the inverse (opposite) of Multiplying.

A negative exponent means how many times** to
divide** by the number.

### Example: **8**^{-1} = 1 ÷ 8 = 1/8 = 0.125

^{-1}= 1 ÷ 8 = 1/8 = 0.125

Or many divides:

### Example: **5**^{-3} = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

^{-3}= 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

**5 ^{-3}** could also be calculated like:

**1 ÷ (5 × 5 × 5) = 1/5 ^{3} = 1/125 = 0.008**

That last example showed an easier way to handle negative exponents:

- Calculate the positive exponent (a
^{n}) - Then take the Reciprocal (i.e. 1/a
^{n})

To change the **sign** (plus to minus, or minus to plus) of the **exponent**,

use the Reciprocal (i.e. 1/a^{n})

So, what about **8 ^{-2}** ?

### Example: **8**^{-2} = 1 ÷ 8 ÷ 8 = 1/8^{2} = 1/64 = 0.015625

^{-2}= 1 ÷ 8 ÷ 8 = 1/8

^{2}= 1/64 = 0.015625

More Examples:

Negative Exponent | Reciprocal of Positive Exponent |
Answer | ||
---|---|---|---|---|

4^{-2} |
= | 1 / 4 ^{2} |
= | 1/16 = 0.0625 |

10^{-3} |
= | 1 / 10 ^{3} |
= | 1/1,000 = 0.001 |

## It All Makes Sense

My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:

Example: Powers of 5 | |||
---|---|---|---|

.. etc.. | |||

5^{2} |
1 × 5 × 5 |
25 | |

5^{1} |
1 × 5 |
5 | |

5^{0} |
1 |
1 | |

5^{-1} |
1 ÷ 5 |
0.2 | |

5^{-2} |
1 ÷ 5 ÷ 5 |
0.04 | |

.. etc.. |

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.