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If 2^x - 2^(x-2) = 3*2^(13), what is x?

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

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If 2^x - 2^(x-2) = 3*2^(13), what is x?

A. 9
B. 11
C. 13
D. 15
E. 17
[Reveal] Spoiler: OA

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Last edited by Bunuel on 25 Jun 2013, 04:17, edited 2 times in total.
Edited the question and added the OA
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Re: 2^x [#permalink]

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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?


Please always post answer choices for PS questions. Question should read:

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

Answer: D.

Hope it's clear.
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2^x – 2^(x-2) = 3(2^13), what is x? [#permalink]

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New post 27 Aug 2013, 22:35
2^x – 2^(x-2) = 3(2^13), what is x?
a. 9
b. 11
c. 13
d. 15
e. 17
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Re: 2^x – 2^(x-2) = 3(2^13), what is x? [#permalink]

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

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New post 28 Aug 2013, 13:27
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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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Re: 2^x [#permalink]

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New post 06 Sep 2013, 10:14
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?


Please always post answer choices for PS questions. Question should read:

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

Answer: D.

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]

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New post 06 Sep 2013, 12:59
theGame001 wrote:
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?


Please always post answer choices for PS questions. Question should read:

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

Answer: D.

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)?


Sure! First notice that \(2^{x-2} * 2^{2} = 2^{x}\) So, we know that \(2^{x-2}\) is a factor of \(2^{x}\). I am using the product rule for exponents: \(x^{a}*x^{b}=x^{a+b}\) It helps to think of this rule in reverse (going from right -> left). What I mean by that is we can also write it as \(x^{a+b}=x^{a}*x^{b}\) When I factor out the \(2^{x-2}\) I am really separating \(2^{x}\) into \(2^{x-2} * 2^{2}\). So,
\(2^x - 2^{x-2} = 3*2^{13}\) which becomes \(2^{x-2}(2^2-1)= 3*2^{13}\) after we factor out the \(2^{x}\).

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

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New post 09 Sep 2013, 08:30
Quote:
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


It seems that in the bolded step above you could have multiplied both sides by 4/3 thus canceling the 3 from the other side out. You would then be left with 4 or (2*2) or 2^2 in addition to the 2^13 leaving you with the 2^15. Just thought that might be a little quicker.
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Re: 2^x [#permalink]

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New post 21 Oct 2013, 06:15
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?


Please always post answer choices for PS questions. Question should read:

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

Answer: D.

Hope it's clear.

Now it's a pretty simple point, but it got silly old me baffled, so i searched up the net and came up with the following explanation.
\(2^{x-2}(2^2-1)= 3*2^{13}\)

If from \(N^x - N^{x-a}, N^{x-a}\) is factored out we will have:
\(N^{x-a} (N^b - 1)\)
Where \(b = x - a\)
Example 1
\(5^8 - 5^5 = 5^5 (5^3 - 1) = 5^5 (125 - 1) = 5^5*124\)
Example 2
\(2^x - 2^{x-2} = 2^x (2^2 - 1) = 2^x (4 - 1) = 2^x . 3\)

Now in Example \(2\) we don't know the value of the \(x\) but we know the difference between\(x\) and\(x-2\) is \(2\) therefore,\(b = 2\).
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Re: 2^x [#permalink]

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New post 26 Feb 2014, 02:09
suk1234 wrote:
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?


Please always post answer choices for PS questions. Question should read:

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

Answer: D.

Hope it's clear.

Now it's a pretty simple point, but it got silly old me baffled, so i searched up the net and came up with the following explanation.
\(2^{x-2}(2^2-1)= 3*2^{13}\)

If from \(N^x - N^{x-a}, N^{x-a}\) is factored out we will have:
\(N^{x-a} (N^b - 1)\)
Where \(b = x - a\)
Example 1
\(5^8 - 5^5 = 5^5 (5^3 - 1) = 5^5 (125 - 1) = 5^5*124\)
[color=#ff0000]Example 2
[b]\(2^x - 2^{x-2} = 2^x (2^2 - 1) = 2^x (4 - 1) = 2^x . 3\)
[/color][/b]

Now in Example \(2\) we don't know the value of the \(x\) but we know the difference between\(x\) and\(x-2\) is \(2\) therefore,\(b = 2\).



The highlighted Example 2 is wrong

2^x - 2^{x-2} =

2^x (1 - 1/4 ) =

2^x . 3/4 =

2^(x-2) . 3 Hope this helps

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

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Re: If 2^x - 2^(x-2) = 3*2^(13), what is x?   [#permalink] 28 May 2016, 23:47
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