Hi gmatpup,
Here is a step by step solution for your benefit.
Assuming 1/2(10)^35 is same as 1/(2*(10^35))
(1/5)^m * (1/4)^18 = 1/2(10)^35
(1/5)^m * (1/2)^(18*2) = 1/2(10)^35
(1/5)^m * (1/2)^(36) = 1/2(10)^35
(1/5)^m * (1/2)^(36) = (1/2)* 1/((5*2)^35)
(1/5)^m * (1/2)^(36) = (1/2)* (1/(5^35) * (1/(2^35))
(1/5)^m * (1/2)^(36) = (1/(2^(35+1)))* (1/((5)^35))
(1/5)^m * (1/2)^(36) = (1/(2^(36)))* (1/((5)^35))
(1/5)^m * (1/2)^(36) = (1/((5)^35))*(1/(2^(35+1)))
(1/5)^m * (1/2)^(36) = (1/((5)^35))*(1/(2^(36)))
(1/5)^m * (1/2)^(36) = (1/5)^35 * (1/2)^(36)
After cancelling the term (1/2)^(36) from both sides(i.e. Multiplying both sides with 2^36), we get
(1/5)^m = (1/5)^35
As bases are equal, powers should also be equal. So, m = 35. Answer is D