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Exponents of Negative Numbers

Squaring Removes Any Negative

When you square a positive or negative number, you always get a positive result (because a negative times a negative gives a positive):

Positive 32 = 3 × 3 = 9
Negative (-3)2 = (-3) × (-3) = 9

"So what?" you say ...

... well take a look at this:

Square root of square

Oh no! When you square a number, then take the square root, you may not end up back at the same number!

You actually 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

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!

So it isn't possible!

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 exponents:

An Even Root of a Negative Number is Not Real

So just be careful when taking square roots, etc.