Question

If you know that the product of two powers is 5 to the power of 3, and that the quotient is 5 to the power of 17, what could the two powers be? Explain how you know this answer is correct.

Answer 1

`5^(10) \cdot 5^(-7) = 5^3`

`(5^(10))/(5^(-7)) = 5^(17)`

Explanation:

You have two equations:

`5^a \cdot 5^b = 5^3`

`(5^a)/(5^b) = 5^(17)`

But since

`x^a \cdot x^b = x^(a+b)`

you only need to solve a+b = 3 and a-b=17

Rewrite the second as a = 17+b and plug it into the first:

17+b + b = 3, then you find, 2b = -14, so b = -7.

Then a=3-b, so a = 10.