Answer: x = -8
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Work Shown:
6^(x-2) = 36^(x+3)
6^(x-2) = ( 6^2 )^(x+3) ... replace 36 with 6^2
6^(x-2) = 6^(2(x+3)) .... use the rule (a^b)^c = a^(b*c)
6^(x-2) = 6^(2x+6) .... distribute
x-2 = 2x+6 .... bases are both 6, so the exponents must be equal
2x+6 = x-2
2x-x = -2-6
x = -8 This is the only solution
Alternatively, you can use logarithms to solve for x. Though in my opinion that is a messier route to take.
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Checking the answer:
Plug x = -8 into the original equation. Simplify both sides. We should get the same value on both sides. If not, then we made an error in solving for x
6^(x-2) = 36^(x+3)
6^(-8-2) = 36^(-8+3)
6^(-10) = 36^(-5)
0.00000001653818 = 0.00000001653818
We get the same value on both sides. This confirms we have the right answer.