Previously we have learned that
3*3*3*3 = 34
which equals 81.
Note that this is 3 to the fourth power. And, in fact, it is 3 to the POSITIVE fourth power.
Also, we learned that:
34 * 33 = 37
we just added the exponents since it was multiplication.
And also
35 /32 = 33
we just subtracted the exponents since it was division.
But consider
35 / 37 = 3-2
Again we subtracted them and got a power of minus 2.
But that looks somewhat strange. What does 3 to the power of negative 2 mean?
We know 3 to the power of 2 means 3*3. Does 3 to the power of negative 2 mean
-3 * -3 ?
Well, no it does not.
To use our above example:
35 / 37
We do know this means (3*3*3*3*3) / (3*3*3*3*3*3*3)
and we also know we could cancel a bunch of 3s and end up with 1/(3*3).
Also, we know that equals 1/32
But above we figured out it was 3-2
Does 1/(3*3) mean the same thing as 3-2 ???
Well, as a matter of fact, YES it does.
In fact, a negative exponent of a base number means "just flip the denominator and numerator" and change the exponent
to the opposite sign. Or said a more "mathematical way", it is the inverse of the same number with an opposite sign on the exponent.
OR, if you would like to hear it in an even more general manner, it is the inverse of the same number with the exponent
multiplied by negative one (-1).
OR, maybe examples would be more helpful.
4-3 = 1/43 that is the inverse with the exponent as positive. So, 4-3 = 1/64
More examples:
3-6 = 1/36
15-3 = 1/153
8-2 = 1/82
Remember, the inverse with the exponent with the opposite sign.
Now, we are going to switch these examples up a little bit.
1/52 = 5-2
1/12-3 = 123
1/8-2 = 82
Now, one more little twist. Say you had a problem which said,
Take the fraction 1/43
and write this with as a fraction with
a denominator of 1. Geeze - apply the rule from above
OR, if you would like to hear it in an even more general manner,
it is the inverse of the same number with the exponent multiplied by negative one (-1).
So, lets try that rule - inverse the number and multiply the exponent by -1. So, what you got??
1/43 = 4-3
which does have a denominator of one.
Here are a few other resources for negative exponents:
The Math Page - Skill In Algebra
Free Math Help
You tube video (10 minutes) Patrick - Just Math Tutor