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

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New post Updated on: 11 Feb 2020, 23:15
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Question Stats:

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

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New post 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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New post 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

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

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New post 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.
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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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New post 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
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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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