A question on indices I am solving a 7th grade text book and came across this question.
$81^x = 1/(125^y)$ where $x$ and $y$ are integers. Find $12xy$.
The answer given is $0$. While I can understand that when $x$ & $y$ are $0$ the above
equation is satisfied, I can't seem to figure how to arrive at the conclusion.
I tried something like:
--> (3^4x)(5^3y) = 1
--> 3^(4x + 3ky) = 1 where k is such that 3^k = 5
--> 4x + 3ky = 0
But with the above am not able to establish that $12xy$ is $0$.
Please help.
Thank you,
Ramana

HINT : 
$$(81)^x(125)^y=1\iff 3^{4x}\cdot 5^{3y}=3^0\cdot 5^0.$$
Notice that both $3$ and $5$ are prime numbers.
The number of each prime number in each side has to be the same.
P.S. A prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself. 
