Math Tutor Bob - Help with math exponents - A number raised to the zeroth power

So, from Exponents #1 we know that

9⁴ * 9³ = 9⁷ and more general a^N * a^M = a^(M+N)

AND

9⁴ / 9³ = 9¹ and more general a^M / a^N = a^(M-N)

Also, we know

(9²)⁴ = (9⁸) and more general (a^M)^N = a^(M*N)

And also, even more basically we know

9⁴= 9*9*9*9 and more general a^N= a*a*a*a ... N times

So, now in this "exponent discussion" let us move on to new information about exponents.

And "moving on" at some point has to include the meaning of a number to the zeroth power.

That is to say something like 9⁰= 1
or more generally a⁰= 1, where "a" is not equal to zero.

Remember that (from above) a^M / a^N = a^(M-N) so,

9⁷ / 9⁴ = 9³

because 7 minus 4 is 3. Easy enough. Lets try

9⁷ / 9⁷

Intuitively we know the answer is 1.
For goodness sakes anything divided by itself is one (1), except zero. Let us apply our rule

a^N / a^M = a^(M-N) then

9⁷ / 9⁷ = 9^7-7 = 9⁰ = 1.

In fact, any number raised to the zeroth power is one (except zero). This is not that tough of a concept.

If you would like a little more information on this subject of a number raised to the zeroth power being 1 or on the subject of zero raised to the zeroth power, then try these:

Home School Math which states " ... zero to zeroth power is usually said to be indeterminate."

FAQ Math Forum

And some mathematicians regard zero raised to the zeroth power to be 1. If you are feeling real frisky then you can read more about this "zero to the zeroth" power here (I warn you that is likely more than you wish to know):

Wikipedia article of "zero to the zeroth power"

And if you even want to read more on this subject (likely more understandable and therefore more entertaining, and the more I read I realize likely good for insomnia):

Ask Mathematician about a zero raised to zeroth power