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:
√(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.