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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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03 Feb 2012, 08:27
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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, 05:17, edited 2 times in total.
Edited the question and added the OA
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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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2^x – 2^(x-2) = 3(2^13), what is x? [#permalink]

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27 Aug 2013, 23:35
2^x – 2^(x-2) = 3(2^13), what is x?
a. 9
b. 11
c. 13
d. 15
e. 17
Math Expert
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Posts: 34092
Followers: 6094

Kudos [?]: 76664 [0], given: 9978

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

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28 Aug 2013, 01:31
Expert's post
rrsnathan wrote:
2^x – 2^(x-2) = 3(2^13), what is x?
a. 9
b. 11
c. 13
d. 15
e. 17

Merging similar topics.
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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
1
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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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06 Sep 2013, 13:59
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?

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)?

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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09 Sep 2013, 09:30
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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21 Oct 2013, 07:15
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?

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.

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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26 Feb 2014, 03:09
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?

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.

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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29 May 2016, 00:47
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Re: If 2^x - 2^(x-2) = 3*2^(13), what is x?   [#permalink] 29 May 2016, 00:47
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