Hey, how's it going? Welcome the the Beat the GMAT community. If you have not done so, you should definitely try searching this question since it has probably been posted often. However, I will provide a solution.

(1/5)^m x (1/4)^ 18 = 1/(2(10^35))

Notice that 1/(2(10^35)) = 1/(2*(2*5)^35)) = 1/(2*(2^35)*(5^35)) = 1/(2^36*5^35)

Also, note that (1/4)^18 = (1/(2^2))^18 = 1/(2^36)

So we have:

(1/5)^m x 1/(2^36) = 1/(2^36) x 1/(5^35)

From comparing both sides, looks like m = 35.

Hope that helps!

Remember some of your exponent properties for this problem:
(x^a)^b = x^(ab)
(x/y)^a = (x^a)/(y^a)

That is awesome! Thank you!

Some of those properties of exponents questions have been killing me.