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If 2^x−2^(x−2)=3∗2^(13) what is the value of x?

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by M7MBA » Fri Apr 20, 2018 2:11 am
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$$If\ \ \ 2^x−2^{\left(x-2\right)}=3\cdot2^{13}\ \ ,\ what\ \ \ is\ \ the\ \ value\ \ of\ \ \ x?$$ A. 9
B. 11
C. 13
D. 15
E. 17

The OA is the option D.

What is the best way to solve this PS question? Should I try option by option? I need some help. Thanks in advanced.
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by Vincen » Fri Apr 20, 2018 2:48 am
Hi M7MBA.

I'd solve this PS question rewriting the expression as follows:

$$2^x-2^{\left(x-2\right)}=3\cdot2^{13}\ \ \ \Rightarrow\ \ \ 2^{x-2}\cdot2^2\ -\ 2^{x-2}=3\cdot2^{13}$$ $$\Rightarrow\ \ 2^{x-2}\left(2^2\ -\ 1\right)=3\cdot2^{13}\ $$ $$\Rightarrow\ \ 2^{x-2}\left(3\right)=3\cdot2^{13}\ $$ $$\Rightarrow\ \ 3\cdot2^{x-2}=3\cdot2^{13}\ $$ $$\Rightarrow\ \ x-2=13$$ $$\Rightarrow\ \ x=15.$$ This way, we get that the correct answer is the option D.

I hope it helps.
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by Jeff@TargetTestPrep » Mon Apr 23, 2018 4:52 pm
M7MBA wrote:$$If\ \ \ 2^x−2^{\left(x-2\right)}=3\cdot2^{13}\ \ ,\ what\ \ \ is\ \ the\ \ value\ \ of\ \ \ x?$$ A. 9
B. 11
C. 13
D. 15
E. 17

From the left side of the equation, we first pull out the common factor 2^(x - 2) from each term. We can then simplify the given equation:

2^(x - 2)[2^2 - 1] = 3 x 2^13

2^(x - 2)[4 - 1] = 3 x 2^13

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

2^(x - 2) = 2^13

x - 2 = 13

x = 15

Answer: D

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by [email protected] » Mon Apr 23, 2018 7:36 pm
Hi All,

While this question might look a bit 'scary', you can solve it with a mix of TESTing THE ANSWERS and Arithmetic.

To start, it's worth noting that the 'right side' of the equation is MORE than 2^13. The 'left side' of the equation is 2^X - (another value), so we know that X MUST be greater than 13. Based on the 'spread' of the Answer choices, the correct answer MUST be either D or E. Since 2^17 is considerably bigger than 2^15, let's TEST Answer D first....

Answer D: 15
IF X = 15.....

2^15 - 2^13..... can be rewritten as...
(2^13)(2^2 - 1) =
(2^13)(3)

This is an exact match for the 'right side' of the equation, so this MUST be the answer.

Final Answer: D

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Rich
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