# Solution for an exponential expression without using logarithms, with two defined variables

If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$?

This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just want to learn it, or it will continue to bug my head

Thanks

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Are you sure it's not 12^((1-a-b)/(2-2b))? Because that works out nicely, but the posted version doesn't. I'll show the working for this version.

60^b = 5, so 12 = 5/60 = 60^(1-b).

So 12^((1-a-b)/(2-2b)) = (60^(1-b))^((1-a-b)/(2-2b))
= 60^((1-a-b)(1-b)/(2-2b))
= 60^((1-a-b)/2)
= (60 / 60^a / 60^b)^(1/2)
= (60 / 3 / 5)^(1/2)
= 4^(1/2)
= 2.

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