Given that (3^(x+2))*(5^(x-1)) = 15^(2x), show that 15^x = 9/5.
I managed to reach: 15^x= [3*sqrt(5)*sqrt(3^x)*sqrt(5^x)]/5, but can't reach 9/5.
I think I misread your question earlier. Apologies. The answer is because taking the square root of both sides is not going to take you anywhere.
There is no method of identifying which technique to use what time; it's just insights that one gains from exposure and experience by doing lots of this.