## Exponents

Table of Contents

## Definition

The exponent of a number says how many times to use the number in a multiplication.

In 4^{2 } the “2” says to use 4 twice in a multiplication,

so

4^{2} = 4 × 4 = 16

In words:

4^{2} could be called “4 to the power 2” or “4 to the second power”, or simply “4 squared”

Exponents are also called Powers or Indices.

Some more examples:

In Numbers | In Words |

3^{3} = 3 × 3 × 3 = 27 | “3 to the third power”, “3 to the power 3”,or simply “3 cubed” |

5^{4} = 5 × 5 × 5 × 5 = 625 | “5 to the fourth power”,”5 to the power 4″, or simply “5 to the 4th” |

Exponents make it easier to write and use many multiplications

7^{6} is easier to write and read than 7 × 7 × 7 × 7 × 7 × 7

We can multiply any number by itself as many times as we want using exponents.

## In General

*a*^{n} tells you to multiply a by itself, so there are n of those *a*‘s:

## If the Exponent is 1 or 0?

Exponent | Answer | Example |

1 | number itself | 8^{1} = 8 |

0 | 1 | 6^{0} = 1^{} |

## What about 0^{0} ?

It could be either 1 or 0, and so people say it is “**indeterminate**“.

## Negative Exponents

What could be the opposite of multiplying?

**Dividing!**

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

4^{-1} = 1 ÷ 4 = 0.25

**You can have many divides**

10^{-2} = 1 ÷ 10 ÷ 10 = 0.01

That can be done in another way:

10^{-2} could also be calculated like:

1 ÷ (10 × 10) = 1/10^{2} = 1/100 = 0.01

Negative Exponent = Flip the Positive Exponent

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

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

More Examples:

Negative Exponent | Reciprocal of Positive Exponent | Answer |

5^{-2}^{} | 1/5^{2}^{} | 1/25 = 0.04 |

2^{-3}^{} | 1/2^{3}^{} | 1/8 = 0.125 |

(-10)^{-3}^{} | 1/(-10)^{3} | 1/(-1000) = -0.001 |

## Makes Sense

Start with “1” and then multiply or divide as many times as the exponent says, then you will get the right answer.

Example

etc… | n^{-2} | n^{-1}^{} | Number (n) | n^{1} | n^{2} | etc… |

… | 2^{-2}=1÷2÷22^{-2}=0.25 | 2^{-1} = 1÷22^{-1} = 0.5 | 2 | 2^{1} = 1×22^{1} = 2 | 2^{2}=1×2×22^{2}=4 | … |

… | 4^{-2}=1÷4÷44^{-2}=0.0625 | 4^{-1} = 1÷44^{-1} = 0.25 | 4 | 4^{1} = 1×44^{1} = 4 | 4^{2}=1×4×44^{2}=16 | … |

… | 5^{-2}=1÷5÷55^{-2}=0.04 | 5^{-1} = 1÷55^{-1} = 0.2 | 5 | 5^{1} = 1×55^{1} = 5 | 5^{2}=1×5×55^{2}=25 | … |

## Another Way of Writing Exponent

Sometimes people use the ^ symbol (above the 6 button on your keyboard), as it is easy to type.

3^2 is the same as 3^{2}

3^2 = 3 × 3 = 9

## About Grouping

To avoid confusion, use parentheses () in cases like this:

With ( ) | Without ( ) |

(-7)^{2} = (-7) × (-7) = 49 | -7^{2} = -(7^{2}) = – (7 × 7) = -49 |

(ab)^{2} = ab × ab | ab^{2} = a × (b)^{2} = a × b × b |