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If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is
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20 Jan 2022, 06:13
20 Jan 2022, 06:13
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Question Stats:
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16% (04:33) wrong based on 6 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
(A) \(\frac{-3}{2}\)
(B) -3
(C) -2
(D) 1
(E) 2
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Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is
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20 Jan 2022, 07:01
20 Jan 2022, 07:01
2
Carcass 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
(A) \(\frac{-3}{2}\)
(B) -3
(C) -2
(D) 1
(E) 2
STRATEGY: As with all GRE Multiple Choice 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 could take longer than testing values.
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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Posts: 1141
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Concentration: General Management
Schools: XLRI Jamshedpur, India - Class of 2014
GMAT 1: 700 Q51 V31
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WE:Engineering (Computer Software)
Re: If 2^(a + 3) = 4^(a + 2) - 48, then the value of a is
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15 Jan 2023, 02:20
15 Jan 2023, 02:20
\(2^{a + 3} = 4^{a + 2} - 48\)
=> \(2^a * 2^3 = 2^{2*(a+2)} - 48\)
=> \(8 * 2^a = 2^{2a+4} - 48\)
=> \(8 * 2^a = 2^{2a}*2^4 - 48\)
=> \(8 * 2^a = 2^{2a}*16 - 8*6\)
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!
Watch the following video to learn the Basics of Exponents
=> \(2^a * 2^3 = 2^{2*(a+2)} - 48\)
=> \(8 * 2^a = 2^{2a+4} - 48\)
=> \(8 * 2^a = 2^{2a}*2^4 - 48\)
=> \(8 * 2^a = 2^{2a}*16 - 8*6\)
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!
Watch the following video to learn the Basics of Exponents
