SOLUTION: How many pairs of positive integers (a,b) satisfy the equation (ab)^a = 46656?

Question 1150788: How many pairs of positive integers (a,b) satisfy the equation (ab)^a = 46656?
Found 2 solutions by greenestamps, MathLover1:
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

46656 = (2^6)(3^6) = 6^6.

We need to write that number in the form (ab)^a.

Because 46656 is the 6th power of a positive integer, the exponent a in (ab)^a must be a factor of 6 -- 6, 3, 2, or 1.

Since the question doesn't ask us to identify the pairs but only asks us how many there are, we could stop here. The answer is that there are four pairs of positive integers (a,b) for which 46656 = (ab)^a.

To get some more useful experience in problem solving, we can continue and find the four pairs.

(1) a = 6: --> b = 1. Solution #1: (a,b) = (6,1)

(2) a = 3: --> b = 12. Solution #2: (a,b) = (3,12)

(3) a = 2: --> b = 108. Solution #3: (a,b) = (2,108)

(4) a = 1: --> b = 46656. Solution #4: (a,b) = (1,46656)

Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

= ±

= ± .............., so =>
= ±
= ±

solutions:
, = ±

answer: pair of positive integers (,)=(,)

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