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 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 same number!
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.
Even Exponents of Negative Numbers
An even exponent always gives a positive (or 0) result.
That simple fact can make our 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 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
If even exponents (such as squaring) never give a negative result, what could x be 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 possible when we 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.