Question

List three different ways to write 5^11 as the product of two powers. Explain why all three of your expressions are equal to 5^11

Answer 1

In multiplying quantities of the same base, the product is simply the base raised to a power that is the sum of the individual powers of the factors. That is,
`(a^n)(a^m)= a^(n+m)`

The term
`5^(11)` can be expressed as the product of two factors with the same base (which is five) and whose powers add up to 11. Three examples are:

`(5^1)(5^(10)) = 5^(1+10) = 5^(11) \\ (5^5)(5^6) = 5^(5+6) = 5^(11) \\ (5^8)(5^3) = 5^(8+3) = 5^(11)`