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# If 2^x-2^(x-2)=3*2^13 what is the value of x?

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If 2^x-2^(x-2)=3*2^13 what is the value of x? [#permalink]

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13 Feb 2010, 11:00
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If $$2^x - 2^{(x-2)} = 3*2^{13}$$), what is the value of x?

A) 9
B) 11
C) 13
D) 15
E) 17

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-2-x-2-x-2-3-2-13-what-is-the-value-of-x-130109.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 14 Apr 2014, 01:15, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: GMAT Prep 1: More Exponents! [#permalink]

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13 Feb 2010, 11:05
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carriedinterest wrote:
If 2^x - 2^(x-2) = 3(2^13), what is the value of x?

A) 9
B) 11
C) 13
D) 15
E) 17

Clear, concise explanations welcome!

D....

2^x - 2^(x-2) = 3(2^13)
2^x - 2^x / 2^2 = 3(2^13)
2^(x+2) - 2^x = 3(2^13)(2^2)
2^x(2^2 -1) = 3(2^15)
2^x(3) = 3(2^15)
Therefore x = 15
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Re: GMAT Prep 1: More Exponents! [#permalink]

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03 Mar 2010, 19:53
Can someone explain how we went from:

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

to

2^x(2^2 -1) = 3(2^15)

?

thanks

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Re: GMAT Prep 1: More Exponents! [#permalink]

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03 Mar 2010, 20:37
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How
2^(x+2) - 2^x
becomes
2^x (2^2 - 1)

Basically you take the equation above, and remove 2^x out of the equation
It's like saying
x^2 - xy
to
x(x-y)

SO it can also be written as
(2^x)(2^2) - (2^x)(1)
which then becomes (2^x)(2^2 - 1)

Hope that was helpful, I'm a GMAT instructor for Manhattan Review.
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Re: GMAT Prep 1: More Exponents! [#permalink]

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03 Mar 2010, 21:10
WillLManhattanReview wrote:
How
2^(x+2) - 2^x
becomes
2^x (2^2 - 1)

Basically you take the equation above, and remove 2^x out of the equation
It's like saying
x^2 - xy
to
x(x-y)

SO it can also be written as
(2^x)(2^2) - (2^x)(1)
which then becomes (2^x)(2^2 - 1)

Hope that was helpful, I'm a GMAT instructor for Manhattan Review.
If you would like help preparing for your GMAT, feel free to look us up.

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A simple factorization! I was confused as i thought (2^x)(2^2 - 1) was (2^x)(2^2-1)

Thank you!

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Re: GMAT Prep 1: More Exponents! [#permalink]

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03 Mar 2010, 21:25
carriedinterest wrote:
If $$2^x - 2^{(x-2)} = 3*2^{13}$$), what is the value of x?

A) 9
B) 11
C) 13
D) 15
E) 17

Clear, concise explanations welcome!

[Reveal] Spoiler:
OA = D

2^{(x-2)}(4-1) = 3*2^{13}
so x-2 = 13
x= 15 hence D.

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Re: GMAT Prep 1: More Exponents! [#permalink]

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03 Nov 2010, 05:50
Simplifying the right side,
3 * 2^13
(2^2 - 1) * 2^13
2^2*2^13 - 2^13 (Multiplying with both terms inside)
2^15 - 2^13 (adding the powers since base is same)

=> 2^x - 2^ (x - 2) = 2^15 - 2^13
=> Hence, x = 15 or x - 2 = 13, x = 15.

HTH
--
Aman

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If 2^x - 2^(x-2) = 3*2^13, what is the value of x? [#permalink]

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14 Nov 2010, 14:34
2
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BOOKMARKED
If $$2^x - 2^{x-2} = 3(2^{13})$$, what value has x?

A. 9
B. 11
C. 13
D. 15
E. 17

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14 Nov 2010, 14:51
Expert's post
1
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BOOKMARKED
Pepe wrote:
Another question I have problem with:

If 2^x - 2^(x-2) = 3*2^13, What is the value of x?
A 9
B 11
C 13
D 15
E 17

$$2^x - 2^{x-2} = 3*2^{13}$$ --> factor out $$2^{x-2}$$ --> $$2^{x-2}*(2^2-1)=3*2^{13}$$ --> $$2^{x-2}*3=3*2^{13}$$ --> $$2^{x-2}=2^{13}$$ --> $$x-2=13$$ --> $$x=15$$.

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15 Nov 2010, 15:18
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Hey guys,

This has always been a favorite problem of mine - the first time I saw it, an instructor had blanked on how to solve it and emailed me a photo from his phone asking for help. He had excused himself from a tutoring session and needed me to explain how to solve it (correctly, of course) within a few minutes, so the pressure was on!

Bunuel's explanation is perfect (as always), but when the pressure was on and I wasn't exactly thinking about factoring, I did this instead - I looked to see if there were a pattern in the subtraction at left (2 to an exponent minus 2 to another exponent, two less) that would always produce 3*something on the right. So I did:

x = 3 and x-2 = 1
2^3 - 2^1 = 8 - 2 = 6

And 6 = 3(2^1), so I had a start.

x = 4 and x-2 = 2
2^4 - 2^2 = 16 - 4 = 12

And 12 = 3(2^2), so the pattern held

x = 5 and x-2 = 3
2^5 - 2^3 = 32 - 8 = 24

And 24 = 3(2^3), and the pattern became clear...
The operation at left was always producing 3*2^(x-2) as its answer, so if x-2 = 13, then x = 15.

Strategically, using small numbers to establish patterns works pretty well when huge numbers (like 3(2^13)) are in play, and when exponents are involved (exponents are essentially just repetitive multiplication, so there are bound to be some repetitive patterns involved). If you can factor like Bunuel did, that's a great way to go...but I'd recommend having the "prove patterns w/ small numers" ideology in your arsenal!
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03 Dec 2010, 10:38
emk60 wrote:
I am having trouble understanding this... I can't remember whose notes this is from, but it is something I printed off this site.

7^4 + 7^7
= 7^4(1+7^3)
=7^4(50)

And

3^4+12^4
=3^4+ (3*4)^4
=3^4(1+4^4)
=3^4(256)

In both instances I am confused with what happens to the one in the second to last step. Wouldn't it be 3^4+3^4(256)? It seems like the 1 just disappears???

Here is a problem I can not solve:

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

x=?

The answer is 15 but I do not understand how to get it. Is there a good guide online or does someone have good notes for exponents? I am lacking in this area and I need it dumbed down.

Provide answer choices for PS questions.
Make sure you type the question in exactly as it was stated from the source.

1. $$7^4+7^7$$ --> factor out 7^4 --> $$7^4(1+7^3)=7^4(1+343)=7^4*344$$ (so the answer given in your example is not correct);

2. $$3^4+12^4=3^4+(3*4)^4=3^4+3^4*4^4$$ --> factor out 3^4 --> $$3^4(1+4^4)=3^4(1+256)=3^4*257$$ (so again the answer given in your example is not correct).

The question itself:
If 2^x - 2^(x-2) = 3*2^13, what is the value of x?
A. 9
B. 11
C. 13
D. 15
E. 17

$$2^x - 2^{x-2} = 3*2^{13}$$ --> factor out $$2^{x-2}$$ --> $$2^{x-2}*(2^2-1)=3*2^{13}$$ --> $$2^{x-2}*3=3*2^{13}$$ --> $$2^{x-2}=2^{13}$$ --> $$x-2=13$$ --> $$x=15$$.

Other solutions at: solving-exponents-ii-104831.html

For more on number theory an exponents check: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Hope it helps.
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03 Feb 2012, 08:44
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Expert's post
manalq8 wrote:
2^x _ 2^x-2=(3)(2^13). what is X?

the way I did this q is

x-x-2=13

but the X's are cancelled so how are we going to get the X?

If 2^x - 2^(x-2) = 3*2^(13), what is x?
A. 9
B. 11
C. 13
D. 15
E. 17

$$2^x - 2^{x-2} = 3*2^{13}$$ --> factor out $$2^{x-2}$$ --> $$2^{x-2}(2^2-1)= 3*2^{13}$$ --> $$2^{x-2}=2^{13}$$ --> $$x-2=13$$ --> $$x=15$$.

Hope it's clear.
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06 Feb 2012, 01:21
2^x-2^(x-2)=3(2^13)
=> 2^(x-2) * [2^2 - 1] = 2^(15-2) * [2^2 - 1]
=> x = 15

Option (4)
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Re: If 2^x - 2^(x-2) = 3*2^13, What is the value of x? [#permalink]

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13 May 2012, 06:27
This is how I solved this problem.

2^x-2^x-2=3(2^13)
Factor 2^x 2^x(1-1)*2^-2=3(2^13)
x-2=13
x=15

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Re: If 2^x - 2^(x-2) = 3*2^13, What is the value of x? [#permalink]

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08 Jun 2012, 21:04
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2^x-2^x-2=3*2^13
2^x(1-2^-2)=2^13*3
2^x(1-1/2^2)=3*2^13
2^x(3/4)=3*2^13
2^x=3*2^13*4/3
2^x=2^13*4
2^x=2^13*2^2
2^x=2^15
Hence, x=15

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Re: If 2^x - 2^(x-2) = 3*2^13, What is the value of x? [#permalink]

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09 Jun 2012, 00:44
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This has the look of not factoring easily, so I applied some logic first.

We can rule out A,B and C (as the left hand side would all be 2^13 less a positive number (smaller than 2^13), so could never equal 3 * 2^13 (larger than 2^13)

Try D:
2^15 - 2^13 = 3*2^13

We can work with this easily:

Divide by 2^13
2^2 - 1 = 3
3 = 3

D

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Re: If 2^x - 2^(x-2) = 3*2^13, What is the value of x? [#permalink]

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09 Jun 2012, 02:39
Hi,

One can always try the options. Values less than 13 can be eliminated. Try others and whenever LHS = RHS, there you get the answer.

Regards,

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22 Jul 2012, 22:43
Hi,

Usually for problems where you have a number at different powers summed you can start by factoring by the smallest power of that number:

$$2^x-2^(x-2)=2^(x-2)*(2^2-1)=2^(x-2)*3$$

Hence

$$x-2=13$$
$$x=15$$

Not sure it is 700+ level

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Re: If 2^x - 2^(x-2) = 3*2^(13), what is x? [#permalink]

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28 Aug 2013, 14:27
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I did it a bit differently, but I arrived at the right answer. Here is the way I did it:
$$2^x - 2^{x-2} = 3* 2^{13}$$
Factor out a $$2^x$$ which gives:
$$2^x(1 - \frac{1}{4}) = 3* 2^{13}$$
Clean up:
$$2^x(\frac{3}{4}) = 3* 2^{13}$$
At this point I realized that the 4 in the denominator could be factored out so that's what I did:
$$2^{x-2}(3) = 3* 2^{13}$$
From here you have $$2^{x-2} = 2^{13}$$ so $$x = 15$$

I hope that helps.
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06 Sep 2013, 11:14
Bunuel wrote:
manalq8 wrote:
2^x _ 2^x-2=(3)(2^13). what is X?

the way I did this q is

x-x-2=13

but the X's are cancelled so how are we going to get the X?

If 2^x - 2^(x-2) = 3*2^(13), what is x?
A. 9
B. 11
C. 13
D. 15
E. 17

$$2^x - 2^{x-2} = 3*2^{13}$$ --> factor out $$2^{x-2}$$ --> $$2^{x-2}(2^2-1)= 3*2^{13}$$ --> $$2^{x-2}=2^{13}$$ --> $$x-2=13$$ --> $$x=15$$.

Hope it's clear.

Can you please explain the red highlighted part? I read other explanations but it wasn't clear. I am unable to understand how did you factor out 2^(x-2)?

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Re: 2^x   [#permalink] 06 Sep 2013, 11:14

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