Question

How do you simplify `3^5*3^4`?

Answer 1

`3^5*3^4=3^9`

Explanation:

Using the property `a^n*a^m=a^(n+m)`:

`3^5*3^4=3^(5+4)=3^9`

To gain some intuition as to why this property works, let's try expanding the exponent into multiplication:

`a^n*a^m=(a*a*a*...*a)(a*a*a*...*a)` where the first parentheses contain `n` `a`'s and the second parentheses contain
`m` `a`'s. Thus, there are `n+m` total `a`'s being multiplied, or `a^(n+m)`

The above is a simple way of seeing why this property works for positive integer exponents, however the property does hold for all exponents (including negative numbers, fractions, irrational numbers, and so on).