# Math Contest Question: Exponents

#### rpk5024

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

405^2 gives 407 as an answer we figured out, but I'm not sure how to show how I got there and it seems too simple to be the best answer.

Thats the question. I was wondering how to do the question ie method and what the answer would be. Any help is appreciated.

#### stapel

##### Super Moderator
Staff member
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.

#### rpk5024

##### New member
Ok thanks, any algebraic method that is purely algebraic would be nice.

ie: disproving all of the other solutions would not be correct (or at least get me a 5/5)