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If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is

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Bunuel
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If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   21 Dec 2021, 02:30
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67% (02:09) correct 33% (02:35) wrong based on 87 sessions
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If \(2^{a + 3} = 4^{a + 2} - 48\), then the value of a is

(A) \(\frac{-3}{2}\)
(B) -3
(C) -2
(D) 1
(E) 2



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D
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BrentGMATPrepNow
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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   21 Dec 2021, 07:34
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Bunuel wrote:
If \(2^{a + 3} = 4^{a + 2} - 48\), then the value of a is

(A) \(\frac{-3}{2}\)
(B) -3
(C) -2
(D) 1
(E) 2

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STRATEGY: As with all GMAT Problem Solving questions, we should immediately ask ourselves, Can I use the answer choices to my advantage?
In this case, all of the answers choices except A are easy to test.
Now we should give ourselves about 20 seconds to identify a faster approach.
In this case, we could try solving the equation, but that doesn't look easy.
So, I'm going to test the answer choices, starting with the easiest ones (which are the two answer choices that are positive integers).

(D) 1
Plug in \(a = 1\), to get: \(2^{1 + 3} = 4^{1 + 2} - 48\)
Simplify: \(16 = 64 - 48\). WORKS!

Answer: D
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ElninoEffect
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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   21 Dec 2021, 05:26
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Bunuel wrote:
If \(2^{a + 3} = 4^{a + 2} - 48\), then the value of a is

(A) \(\frac{-3}{2}\)
(B) -3
(C) -3
(D) 1
(E) 2



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Bunuel both options B and C are -3, I think one should have been +3. Not nitpicking just a humble observation and suggestion. :)
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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   21 Dec 2021, 05:29
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ElninoEffect wrote:
Bunuel wrote:
If \(2^{a + 3} = 4^{a + 2} - 48\), then the value of a is

(A) \(\frac{-3}{2}\)
(B) -3
(C) -3
(D) 1
(E) 2



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Bunuel both options B and C are -3, I think one should have been +3. Not nitpicking just a humble observation and suggestion. :)

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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   23 Dec 2021, 00:54
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The Algebra approach
4^a+2-2^a+3=2^4*3
2^2(a+2)-2^(a+3)=2^4*3
2^2a*2^4-2^a*2^3=2^4*3
2^3*2^a[2*2^a-1]=2^4*3
Equating the powers
a+3=4, a=1
Answer D
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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   31 Jan 2022, 05:38
2^a+3 = 4^a+2 - 48
4^a+2 - 2^a+3 = 48
2^2a+4 - 2^a+3 = 2^4*3
2^3*2^a(2^a-1) = 2^4*3
2^a(2^a-1) = 2*3

3 can be also be written as = (2^2-1).

Therefore,
2^a(2^a-1) = 2*(2^2-1)

a=1 or a-1=2 => a=1 (C)
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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink] New post   15 Jan 2023, 03:21
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\(2^{a + 3} = 4^{a + 2} - 48\)
=> \(2^a * 2^3 = 2^{2*(a+2)} - 48\)
=> \(8 * 2^a = 2^{2a+4} - 48\)
=> \(8 * 2^a = 2^{2a}*2^4 - 48\)
=> \(8 * 2^a = 2^{2a}*16 - 8*6\)

Dividing both the sides by 8 we get
=> \(2^a = (2^a)^2*2 - 6\)

Let \(2^a\) = x
=> x = 2*\(x^2\) - 6
=> 2*\(x^2\) - x - 6 = 0
=> 2*\(x^2\) - 4x + 3x - 6 = 0
=> 2x*(x - 2) + 3*(x - 2) = 0
=> (x - 2) * (2x + 3) + 0
=> x = 2, \(\frac{-3}{2}\)

=> \(2^a\) = 2 or \(2^a\) = \(\frac{-3}{2}\)
But \(2^a\) cannot be negative
=> \(2^a\) = 2 = \(2^1\)
=> a = 1

So, Answer will be D
Hope it helps!

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Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is [#permalink]
15 Jan 2023, 03:21
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