# Exponents of Negative Numbers

## Squaring Removes Any Negative

"Squaring" means to multiply a number by itself.

When you square a **positive** number you get a positive result. Example: (+5) × (+5) = +25

When you square a **negative** number, you also get a **positive** result: (-5) × (-5) = +25 (because a negative times a negative gives a positive):

*"So what?"* you say ...

... well take a look at this:

Oh no! We started with **minus 3** and ended with **plus 3**.

When you **square** a number, then take the **square root**, you may not end up with the same number!

In fact you end up with the absolute value of the number:

**√(x^{2})** =

**|x|**

That also happens for all even (but not odd) Exponents.

## Even Exponents of Negative Numbers

An even exponent always gives a **positive** (or 0) result.

That simple fact can make your life easier:

1 (Odd): | (-1)^{1} = -1 |

2 (Even): | (-1)^{2} = (-1) × (-1) = +1 |

3 (Odd): | (-1)^{3} = (-1) × (-1) × (-1) = -1 |

4 (Even): | (-1)^{4} = (-1) × (-1) × (-1) × (-1) = +1 |

Do you see the -1, +1, -1, +1 pattern?

(-1)^{odd} =** -1**

(-1)^{even} =** +1**

So you can "shortcut" some calculations, like:

### Example: What is (-1)^{97} ?

97 is odd, so:

(-1)^{97} = -1

### Example: What is (-2)^{6} ?

2^{6} = 64, and 6 is even, so:

(-2)^{6} = +64

## Roots of Negative Numbers

If even exponents (such as squaring) never give a negative result, what could x be here:

x^{2} = -1

**Does x=1?**

1 × 1 = **+1**

**Does x=-1?**

(-1) × (-1) = **+1**

We can't get -1 for an answer!

**It seems impossible!**

Well, it **is** possible when you use Imaginary Numbers. But not with Real Numbers.

In other words:

√-1 is not a Real Number

This is true for all even roots:

An Even Root of a Negative Number is Not Real

So just be careful when taking square roots, 4th roots, etc.