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:
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.
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