rpk5024 said:

3^8 • 5^2 = a^b , where both a and b are positive integers, find the smallest possible value for a + b.

405^2 =407 we figured out

But 405<sup>2</sup> = (405)(405) = 164025, not anywhere close to 407 (which is just 405

*plus* 2, not

*raised to the power* 2).

Do you perhaps mean "a = 405, b = 2, so a<sup>b</sup> = 405<sup>2</sup>, and a + b = 405 + 2 = 407"...?

Since the prime factor 5 occurs only twice, any power on a base containing 5 has a factor can be no greater than 2, since any larger power would return too many copies of the factor 5. So, from an algebraic standpoint, the solution "a = 405, b = 2" would seem the best possible.

Eliz.