The remainder of a product is a product of the remainders of each factor.

Break down into large factors:
2^39 = 2^9 * 2^9 * 2^9 * 2*9 * 8

Calculate remainder of each factor:
2^9 = 512, divided by 39 gives a remainder of 5
8 divided by 39 gives a remainder of 8

Multiply the remainders of the 5 factors and try to reduce to a remainder less than 39 by subtracting multiples of 39:
5*5*5*5*8 = 125*40 = 125*(39+1) = 125*39 + 125 = 125*39 + 3*39 + 8

So the reduced remainder is 8.

In general, such a laborous problem won't appear on GMAT, unless there is a faster shortcut.

Anurag i feel that the modular method stated here consumes the same time as the previous method.
I doubt such a hardworking problem is asked in the GMAT as it will surely eat away a lot of time!!!
Kindly conform if there is possibility of such question. : :

They are the same method only mine is explained without formulas.