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IF 2^x 2^(x-2) = 3(2^13), what is the value of x? 9 11 13 15

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Manager
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IF 2^x 2^(x-2) = 3(2^13), what is the value of x? 9 11 13 15 [#permalink] New post 05 Jan 2008, 18:57
IF 2^x – 2^(x-2) = 3(2^13), what is the value of x?

9
11
13
15
17

I guessed and got this one right, but have no idea how to solve. Please Help!
SVP
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Re: Another exponent question from GMATprep Test 1 [#permalink] New post 05 Jan 2008, 19:38
i get x=15

i start off by factoring out 2^x from the right side, which leaves you with:

2^x(1-2^-2)
2^x(1-1/4)
2^x(3/4)

3/4 can be written as 3/2^2, and plugging in above you get:

2^x(3/2^2) --> (3)(2^x-2) = 3(2^13), which is given in the stem

2^x-2 = 2^13, i.e. x-2=13, x=15
Director
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Re: Another exponent question from GMATprep Test 1 [#permalink] New post 05 Jan 2008, 19:51
wow! what a question! I haven't seen this one before, but after a little playing around with it here's what I've come up with:

2^(x-2) will always be 1/4 of 2^x. This is because you're taking 2*2 (or 4) out of the equation (with the -2). The result will always be missing the 2*2 so it will be 1/4 of 2^x. Here are some examples to demonstrate:

2^3 = 2*2*2 = 8
2^1 = 2 = 2
8/2 = 4

2^5 = 2*2*2*2*2 = 32
2^3 = 2*2*2 = 8
32/8 = 4

2^8 = 2*2*2*2*2*2*2*2 = 256
2^6 = 2*2*2*2*2*2 = 64
256/64 = 4

so whatever number you get with 2^x, 2^(x-2) will always end up as 1/4 of that.

Now what does this mean for us? This means that

2^x - 2^(x-2) is equal to x - 1/4x or (3/4x) as a total
if 2^(x-2) is 1/4x and the total is 3/4x then you can say that 2^x-2^(x-2) = 3(2^x-2)

Now we have:

3(2^(x-2)) = 3(2^13)
x-2 = 13
x = 15

Answer D

Hope that helps! Let me know if my explanation leaves something to be desired :)
SVP
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Re: Another exponent question from GMATprep Test 1 [#permalink] New post 05 Jan 2008, 20:12
eschn3am wrote:
wow! what a question! I haven't seen this one before, but after a little playing around with it here's what I've come up with:

2^(x-2) will always be 1/4 of 2^x. This is because you're taking 2*2 (or 4) out of the equation (with the -2). The result will always be missing the 2*2 so it will be 1/4 of 2^x. Here are some examples to demonstrate:

2^3 = 2*2*2 = 8
2^1 = 2 = 2
8/2 = 4

2^5 = 2*2*2*2*2 = 32
2^3 = 2*2*2 = 8
32/8 = 4

2^8 = 2*2*2*2*2*2*2*2 = 256
2^6 = 2*2*2*2*2*2 = 64
256/64 = 4

so whatever number you get with 2^x, 2^(x-2) will always end up as 1/4 of that.

Now what does this mean for us? This means that

2^x - 2^(x-2) is equal to x - 1/4x or (3/4x) as a total
if 2^(x-2) is 1/4x and the total is 3/4x then you can say that 2^x-2^(x-2) = 3(2^x-2)

Now we have:

3(2^(x-2)) = 3(2^13)
x-2 = 13
x = 15

Answer D

Hope that helps! Let me know if my explanation leaves something to be desired :)


how much time you need to solve this?
Director
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Re: Another exponent question from GMATprep Test 1 [#permalink] New post 05 Jan 2008, 20:15
The hardest part is knowing what to look for. If you can catch the reasoning behind 2^x-2 being 1/4 of 2^x right off the bat the rest of it should fall into place in less than a minute.
Re: Another exponent question from GMATprep Test 1   [#permalink] 05 Jan 2008, 20:15
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IF 2^x 2^(x-2) = 3(2^13), what is the value of x? 9 11 13 15

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