# Exponents of Negative Numbers

## Squaring Removes Any Negative

"Squaring" means to multiply a number by itself.

- Squaring a
**positive**number gets a**positive**result: (+5) × (+5) = +25 - Squaring a
**negative**number also gets a**positive**result: (−5) × (−5) = +25

Because a negative times a negative gives a positive. So:

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

... well take a look at this:

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

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

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

**√(x^{2})** =

**|x|**

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

Try here:

## Even Exponents of Negative Numbers

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

That simple fact can make our life easier:

^{1}=

**−1**

^{2}= (−1) × (−1) =

**+1**

^{3}= (−1) × (−1) × (−1) =

**−1**

^{4}= (−1) × (−1) × (−1) × (−1) =

**+1**

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

(−1)^{odd} =** −1**

(−1)^{even} =** +1**

So we 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

### Example: What is the value of x 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** impossible using Real Numbers.

But we **can** do it using Imaginary Numbers.

In other words:

√−1 is **not** a Real Number ...

... it is an Imaginary 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, 6th roots, etc.