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:
$(5^1)(5^{10}) = 5^{1+10} = 5^{11} \\ (5^5)(5^6) = 5^{5+6} = 5^{11} \\ (5^8)(5^3) = 5^{8+3} = 5^{11}$