Which of the following is equal to 2^5+2^5+3^5+3^5+3^5 ?

[itheenigma]

GMAT Prep problem.

Which is equal to 2^5+2^5+3^5+3^5+3^5?
a. 5^6
b. 13^5
c. 2^6 + 3^6
d. 2^7 + 3^8
e. 4^5 +9^5

I got the correct answer ( C ), but was not able to think up of an elegant way to do this.
My method was brute force calculation.
I'm sure there's a more graceful way to do this?

[sam2304]

2^5 + 2^5 + 3^5 + 3^5 + 3^5
= 2^5(1+1) + 3^5(1+1+1)
2^5*2 + 3^5 * 3
2^6 + 3^6.

IMO C.

Hope this helps.

[itheenigma]

Doh!
The brain sometimes refuses to work...

Thanks though!

[melguy]

Thanks

What if we change the question a bit and have a larger number. How will we approach this problem

e.g.

2^180 - 2^30

Do we take 2^30 common
2^30 (2^150 - 1)
2^30 (2^150 - 2^0)

I get stuck here. Please help.

[neelgandham]

Thanks

What if we change the question a bit and have a larger number. How will we approach this problem

e.g.

2^180 - 2^30

Do we take 2^30 common
2^30 (2^150 - 1)
2^30 (2^150 - 2^0)

I get stuck here. Please help.

Solving exponent problems is easy irrespective of the magnitude of the exponent/s. GMAT doesn't test your mathematical ability in solving an exponent problem but tests your logical ability in solving(just my experience). During the initial phase of preparation, I will advise you to have a quick look at the options before you start solving it. Let the question be framed as below.

The expression 2^180 - 2^30 is same as ?

a) 2^30 *(2^151 - 1)
b) 2^31 *(2^150 - 2^0)
c) 2^106 *((2^75)-(1/2^75))
d) 2^106 *((2^75)-(1/2^75))
e) 2^105 *((8^25)-(1/8^25))

After a quick look at the answer options, one can categorize the options into three different types of the form 2^a * (2^b - 1),2^a * (2^b - 2^c),2^a * (8^b - 8^c). Then simplify the expression to a point where you can compare it with one of the above categories. After you are used to solving exponent problems, you got to think the GMAT way !

GMAT way - Start with C(just my way) and expand the expression

c) 2^106 *((2^75)-(1/2^75)) = 2^(106+75) - 2^(106-75) - Not the answer, now to option D
d) 2^105 *((2^76)-(1/2^76)) = 2^(105+76) - 2^(105-76) - Not the answer, now to option E
e) 2^105 *((8^25)-(1/8^25)) = 2^(105+75) - 2^(105-75) - correct ! Answer: Option E

[rooster]

Think of it like this:

A^5 + A^5 + B^5 + B^5 + B^5

Even though we are dealing with exponents, the exponent does not change. Therefore, we get this as a result:

2(A^5) + 3(B^5)

In any other scenario, that is it. But, for this equation, A happens to be 2, and B = 3.

2(2^5) is the same as saying 2(2*2*2*2*2), which is just 2*2*2*2*2*2. That is why it can be 2^6 and 3^6.

[melguy]

.........

Thanks for your input but it flew 100 mtrs above my head lol (i am a science student and maths is my weak spot). But i get the idea that if we are trying to subtract 2^30 from 2^180 then it wont make any huge difference and the answer will stay closer to 2^180 vs climbing at 2^150.

Also, thanks to others for ur contribution!