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):

5x5 = -5x-5

"So what?" you say ...

... well take a look at this:

Square root of square

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:

√(x2) = |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 ?

26 = 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:

x2 = -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.