Hi Sheila,
The reason that mathematicians use the convention that a base number raised to the power of 0 be 1 comes from an attempt to be consistent with the notation. There are some nice properties that come from our use of exponential notation. For example
5^{3} 5^{4} = (5 5 5) (5 5 5 5) = 5^{7}
so you can simplify 5^{3} 5^{4} by just adding the exponents, 3 + 4 = 7.
Likewise if you use negative exponents
5^{3} 5^{4} = (5 5 5) ( ^{1}/_{5} ^{1}/_{5} ^{1}/_{5} ^{1}/_{5}) = ^{1}/_{5} = 5^{1}
Again 5^{3} 5^{4} can be simplifies by just adding the exponents, 3 + (4) = 1.
So what about 5^{3} 5^{3}? We know that
5^{3} 5^{3} = (5 5 5) ( ^{1}/_{5} ^{1}/_{5} ^{1}/_{5}) = 1
and yet the addition of the exponents gives 3 + (3) = 0. Hence if we are going to be consistent with the notation it makes sense to say
5^{0} = 1.
It's not a "fact" like 1 + 1 = 2 but rather a convention that mathematicians use so that the simplification by adding exponents is true, even when the sum is 0.
I hope this helps,
Penny
