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# If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?

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Manager
Joined: 29 Oct 2019
Posts: 221
If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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Updated on: 11 Feb 2020, 23:15
5
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Difficulty:

55% (hard)

Question Stats:

56% (01:39) correct 44% (02:28) wrong based on 62 sessions

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If $$(2^x + 8)^2 - (2^x - 8)^2 = 2^{12}$$, what is the value of x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Source: GMAT Quantum

Originally posted by sjuniv32 on 11 Feb 2020, 19:22.
Last edited by Bunuel on 11 Feb 2020, 23:15, edited 1 time in total.
Renamed the topic and edited the question.
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Posts: 355
Location: United States
Re: If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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11 Feb 2020, 19:44
1
sjuniv32 wrote:
If $$(2^x + 8)^2 - (2^x - 8)^2 = 2^{12}$$, what is the value of x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Source: GMAT Quantum

Explanation:
(2^x + 8)^2 - (2^x - 8)^2 = 2^12
a^2 - b^2 = (a+b)X(a-b)
(2^x + 8 + 2^x - 8)X(2^x + 8 - 2^x + 8)=2^12
{2^(x+1)}X{16} = 2^12
2^(x+1) = 2^8
x+1=8
x=7

IMO-B
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Re: If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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16 Feb 2020, 05:47
2
sjuniv32 wrote:
If $$(2^x + 8)^2 - (2^x - 8)^2 = 2^{12}$$, what is the value of x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Source: GMAT Quantum

We need to recognize that the left hand side of the equation is a difference of two squares; therefore, it can be factored as:

[2^x + 8 - (2^x - 8)][2^x + 8 + (2^x - 8)] = 2^12

(16)(2 * 2^x) = 2^12

(2^4)(2^(x + 1)) = 2^12

2^(x + 5) = 2^12

x + 5 = 12

x = 7

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If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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16 Feb 2020, 11:35
1
sjuniv32 wrote:
If $$(2^x + 8)^2 - (2^x - 8)^2 = 2^{12}$$, what is the value of x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Source: GMAT Quantum

Note the left side is a difference, and in particular a difference of squares while the right side is a single term. To relate two terms and a product, we can use our mighty tool of factoring! In this case, since the left side is a difference of squares we can use a typical formula, $$x^2 - a^2 = (x + a)(x - a)$$. Here $$x = 2^x + 8$$ and $$a = 2^x - 8$$.

Then we have:

$$(2^x + 8)^2 - (2^x - 8)^2 = (2^x + 8 + 2^x - 8)(2^x + 8 - 2^x - (-8)) = (2*2^x)(16) = (2^{x + 1})*2^4 = 2^{x + 5}$$

Since that is equal to $$2^{12}$$ we have $$x + 5 = 12$$ and $$x = 7$$.

Ans: B
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Manager
Joined: 29 Oct 2019
Posts: 221
Re: If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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08 Mar 2020, 21:25
rajatchopra1994 wrote:
sjuniv32 wrote:
If $$(2^x + 8)^2 - (2^x - 8)^2 = 2^{12}$$, what is the value of x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Source: GMAT Quantum

Explanation:
(2^x + 8)^2 - (2^x - 8)^2 = 2^12
a^2 - b^2 = (a+b)X(a-b)
(2^x + 8 + 2^x - 8)X(2^x + 8 - 2^x + 8)=2^12
{2^(x+1)}X{16} = 2^12
2^(x+1) = 2^8
x+1=8
x=7

IMO-B

Thanks a lot for making the problem easier, and this line ''a^2 - b^2 = (a+b)X(a-b)'' helps to understand the problem.
Manager
Joined: 29 Oct 2019
Posts: 221
Re: If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?  [#permalink]

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10 Mar 2020, 18:54
1
Let’s (2^x + 8)^2 = a. And (2^x - 8)^2 = b

So a^2 – b^2= 2^{12}

(2^x + 8 + 2^x - 8) (2^x + 8 - 2^x + 8) = 2^{12}

(2. 2^x) (2^4) = 2^{12}

2^x = 2^{12}/2^5

2^x = 2^7

X = 7
Re: If (2^x + 8)^2 - (2^x - 8)^2 = 2^12, what is the value of x?   [#permalink] 10 Mar 2020, 18:54
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