SOLUTION: why is 3 to the zero power equals 1?

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Question 176744: why is 3 to the zero power equals 1?
Found 2 solutions by Mathtut, MathTherapy:
Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!

ok we know that n%5Ex%2Fn%5Ey=n%5E%28x-y%29
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now take a look at
:
some specifics 3%5E4%2F3%5E2=3%5E%284-2%29=3%5E2
................3%5E4%2F3%5E3=3%5E%284-3%29=3%5E1
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and now our special case
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3%5E4%2F3%5E4=3%5E%284-4%29=3%5E0
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but remember 3%5E4%2F3%5E4=1
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therefore any number raised to the zero power = 1


Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
If we divide 10 by 10 or x by x, or 100 by 100, or any expression by itself, the result will always be 1 (anything divided by itself equals 1).

Based on the Law of Exponents, when dividing expressions with the same base, we need to subtract the exponent in the denominator from the exponent in the numerator.

All expressions without any exponents actually have an exponent of 1. For example, 10 is actually 10%5E1, x is x%5E1, 100 is 100%5E1, and so on. When we divide any of these expressions, such as 10%5E1 by itself, we’re supposed to get a result of 1, just like when we divide 10%5E3 by itself. This is because anything divided by itself equals 1 (see above). Furthermore, based on the Law of Exponents, and as stated above, 10%5E3 divided by 10%5E3 equals 10%5E%283-3%29 which equals 10%5E0. Did we not say before that 10%5E3 ÷ 10%5E3, or anything divided by itself equals 1? Therefore, it follows that:
10%5E3 ÷ 10%5E3 = 1. Also, 10%5E3 ÷ 10%5E3 = 10%5E%283-3%29 which equals 10%5E0.
This means that 10%5E3 ÷ 10%5E3 = 10%5E%283-3%29 = 10%5E0 = 1.
I hope you now understand why any expression to the power of 0 (zero) always equals 1.