Exponentiation with positive integer exponents is analogous to multiplication. `a*3=a+a+a` shows that multiplication by a positive integer is repeated addition. `a^3=a*a*a` shows that exponentiation with a positive integer exponent is repeated multiplication.

If we define `a^n=a*a*...*a{n"times"}` then we can develop other rules.

`a^n*a^m=(a*a*...*a{n"times"})(a*a*...*a{m"times"})`

`=(a*a*...*a{n+m"times"})=a^(n+m)`

Ex: `3^3*3^4=(3*3*3)(3*3*3*3)=3^7=3^(3+4)`

`(a^n)^m=a^n*a^n*...*a^n{m"times"}=a^(nm)` ``` `

`(ab)^n=(a*...*a{n"times"})(b*...*b{n"times"})`

`=a^nb^n`

If we write a sequence of descending powers of a, we note that we get the successive elements by dividing the previous element by a: for example writing powers of 3 in descending order we get 81,27,9,3,... Each term is the previous term divided by 3.

`3^4,3^3,3^2,3^1,3^0=1` makes sense since to get the term with exponent 0 we divide the term with exponent 1 by 3: 3/3=1.

Continuing we see that `3^(-1)=1/3,3^(-2)=1/9,...` since we keep dividing by 3. Thus we are led to the rules :

`a^0=1,a!=0`

`a^(-n)=1/a^n,a!=0`

These rules should also hold for rational numbers, so:

`(a/b)^n=a/b*a/b*...*a/b{n"times"}=a^n/b^n`

`a^n/a^m=a^n*1/a^m=a^n*a^(-m)=a^(n-m)`

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