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If 2x+2x+2x+2x=2a, x=?

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If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 04:15
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If \(2^x+2^x+2^x+2^x=2^a\), x=?

A. a-2
B. a+2
C. 2-a
D. 2a
E. a-4

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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 06:56
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MathRevolution wrote:
If \(2^x+2^x+2^x+2^x=2^a\), x=?

A. a-2
B. a+2
C. 2-a
D. 2a
E. a-4


Nice question!

We need to recognize here that we first need to simplify the left side of the equation.
We are adding 2^x FOUR times, which is the same as (4)2^x
So...

2^x + 2^x + 2^x + 2^x = 2^a
(4)(2^x) = 2^a
(2^2)(2^x) = 2^a [I rewrote 4 as 2^2]
2^(x+2) = 2^a [applied the product law for exponents]

Since we have the same base of 2 on each side of the equation, we can conclude that the exponents are equal. So....
x+2 = a
Subtract 2 from both sides to get: x = a-2

Answer:

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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 11:36
MathRevolution wrote:
If \(2^x+2^x+2^x+2^x=2^a\), x=?

A. a-2
B. a+2
C. 2-a
D. 2a
E. a-4



substitute x=2 in 2^x+2^x+2^x+2^x
2^x+2^x+2^x+2^x= 16 = 2^4 = 2^a
thus a=4 & x=2

check with options which satisfies x=2 when a=4
only option A satisfies x= 2

Hence Ans A
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 12:08
4*(2^x) = (2^a) => (2^2)X(2^x) = 2^a
=> 2^(x+2) = 2^a
=> x+2 = a
=> x = a-2
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 21:14
There are FOUR 2s with power X, so they would add to make
4*(2^)x = (2)^a
since 4=2^2
Therefore, (2)^2 *(2)^x = (2)^a
which results in;
(2)^(2+x) = (2)^a
=> x+2=a
x=a-2
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If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 25 Jan 2017, 22:50
easy one, and thank you, GMATPrepNow :)
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 27 Jan 2017, 03:21
==> From \(2^x+2^x+2^x+2^x=(1+1+1+1)2^x=4(2^x)=2^22^x=2^2^+^x=2^a\), you get \(2+x=a, x=a-2\).

Therefore, the answer is A.
Answer: A
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 30 Jan 2017, 16:58
MathRevolution wrote:
If \(2^x+2^x+2^x+2^x=2^a\), x=?

A. a-2
B. a+2
C. 2-a
D. 2a
E. a-4


Keep in mind that in the first step we are ADDING the four terms on the left.

2^x + 2^x + 2^x + 2^x = 2^a

4(2^x) = 2^a

(2^2)(2^x) = 2^a

2^(2 + x) = 2^a

Because we have equal bases, we can equate the exponents:

2 + x = a

x = a - 2

Answer: A
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 31 Jan 2017, 15:09
2^x ( 4 ) = 2^a
2^x 2^2 = 2^a
2^x+2 = 2^a
x + 2 = a
x = a - 2
A
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Re: If 2x+2x+2x+2x=2a, x=?  [#permalink]

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New post 19 Jan 2019, 22:01
MathRevolution wrote:
If \(2^x+2^x+2^x+2^x=2^a\), x=?

A. a-2
B. a+2
C. 2-a
D. 2a
E. a-4


\(2^x+2^x+2^x+2^x=2^a\)

4 *\(2^x = 2^ a\)

\(2^{x+2} = 2^a\)

x = a-2

Answer A
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Re: If 2x+2x+2x+2x=2a, x=?   [#permalink] 19 Jan 2019, 22:01
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