Easier way to find powers

in this type of problems where you have large exponants, you need to get the comman factor(in another words, factor exponantial terms that have the same base) out and this is how you could simplify it. you factor as how you can do it with equations.
for the 1st example:

3^180 - 3^30=
3^30(3^150-1)

for the second example:

3^180-3^30 + 2^120-2^60

the common factor for the first part is 3^30

3^30(3^150-1)

and for the second part, the common factor is 2^60

2^60(2^60-1)

3^30(3^150-1)+2^60(2^60-1)

hope this helps, please do not hesitate to ask further questions

thanks for clarifying. I too was thinking along these lines.

Lets take the easier Q .
3^180 - 3^30=
3^30(3^150-1)

How can we solve this thing to get a clean answer. I remember 200% the answer choices were in powers (something like below and not big huge numbers) .

a. 3^180
b 3^150
c 3^30

If we calculate this -> 3^30(3^150-1) .. i think there is no easy solution to the second part - (3^150-1). this is the puzzling part to me. How do we quickly solve this thing - 3^30(3^150-1).

There is no 'simple' way to write 3^180 - 3^30 as a single power. All you can do is factor it; you can first factor out 3^30, then if you want to go further you can use a difference of squares:

3^180 - 3^30 = 3^30 (3^150 - 1) = 3^30 (3^75 + 1)(3^75 - 1)

To factor that any further, you'd need to know some factorization patterns that aren't tested on the GMAT. Notice that by factoring, we aren't making the answer appear any simpler - in fact, it begins to look more complicated.

So if you're '200% sure' that you saw a similar question in which the answer choices were neat powers for a similar question, you'd need to show us the exact question. The only similar question I could imagine where the answers would be simple powers would be something like:

3^180 - 3^30 is closest to which of the following:

A) 3^180
B) 3^177
C) 3^150
D) 3^30
E) 3^6

This question is not asking for a simplification at all; it's asking for an estimate. Here you simply need to notice that 3^180 is absolutely enormous in comparison to 3^30, so subtracting 3^30 will barely make any difference at all, and the answer is A.

Thanks a lot for making it clear!