SOLUTION: How many pairs of integers (a,b) satisfy the equation ab^a = 648?

Question 1209333: How many pairs of integers (a,b) satisfy the equation ab^a = 648?
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Find the prime factorization of 648:

648 = (2^3)(3^4)

The exponents in the prime factorization are 3 and 4. Since we want to write 648 in the form ab^a, a can be at most 3. Note a negative would make ab^a a negative fraction, and a=0 would make ab^a equal to zero. So a can only be 1, 2, or 3.

Try each one to see if it fits the pattern.

(1) a=1
1(b^1) = 648
b^1 = 648
b = 648
solution #1: (a,b) = (1,648)

(2) a=2
2(b^2) = 648
b^2 = 324
b = 18 or b = -18
solutions #2 and 3: (a,b) = (2,18) and (a,b) = (2,-18)

(3) a=3
3(b^3) = 648
b^3 = 216
b = 6
solution #4: (a,b) = (3,6)

ANSWER: 4 pairs of integers (a,b) satisfy ab^a = 648

RELATED QUESTIONS
How many pairs of integers (a,b) satisfy the equation {{{ab^a}}} = 648? CC11F... (answered by greenestamps)

How many pairs of positive integers (a,b) satisfy the equation (ab)^a =... (answered by greenestamps,MathLover1)

How many pairs of integers (a,b) satisfy the equation {{{a^ab =... (answered by ikleyn)

How many pairs of integers (a, b) satisfy the equation {{{ a^b =1296... (answered by ikleyn,Boreal)

How many ordered pairs of positive integers (a,b) satisfy the following equation? 2a + 3b (answered by LinnW)

How many pairs of integers (b,c) satisfy the equation... (answered by robertb)

How many pairs of integers (b,c) satisfy the equation: (b+7)/(b+4)=c/9 (answered by Edwin McCravy)

How many ordered pairs (a,b) of positive integers satisfy a^2 + b^2 = 50 (answered by vleith)

How many ordered triples of integers (a, b, c) satisfy |a+b|+c=19 and ab+|c|= 97... (answered by ikleyn)