It helps to go back to how exponents first come up: as an abbreviation for lots of multiplications. We abbreviate a whole bunch of things in math - multiplication first comes up as an abbreviation for repeated addition; we could write out 3 + 3 + 3 + 3 + 3 + 3 every time we want to say "3, six times", but it's much more convenient to write it as "3 x 6" and save yourself the paper space and pencil lead.
Exponentiation comes as the next step up; when we've got something like 3 x 3 x 3 x 3 x 3 x 3 - 6 3's being multiplied together - we can abbreviate it with the exponent [tex] 3^6 [/tex] instead of writing out all of those individual 3's and multiplication signs. Here are the first few "powers" of 3 and their abbreviations:
[tex] 3=3^1=3\\
3\times3=3^2=9\\
3\times3\times3=3^3=27\\
3\times3\times3\times3=3^4=81 [/tex]
Notice that each time the exponent goes up, our number gets multiplied by 3. What if we went in the other direction? We'd be doing the exact opposite thing as the exponents went down - we'd be dividing by 3 each time.
[tex] 3^3=27\\3^2=27\div3=9\\3^1=9\div3=3 [/tex]
And we can continue this pattern one more time to find that
[tex] 3^0=3\div3=1 [/tex]
There's nothing special about our choice of number, by the way. We could say the same about 5:
[tex] 5^2=25\\
5^1=25\div5=5\\
5^0=5\div5=1 [/tex]
or 7:
[tex] 7^2=49\\7^1=49\div7=7\\7^0=7\div7=1 [/tex]
Or any number we want - any number raised to the 0 power will always give us 1, because any number divided by itself is 1.
