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Which of the following is equal to..for all integers x and y

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Which of the following is equal to..for all integers x and y [#permalink] New post 20 May 2016, 17:51
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Question Stats:

84% (01:05) correct 15% (01:26) wrong based on 33 sessions
Which of the following is equal to \(\frac{2^{x-y}}{2^{x+y}}\) for all integers \(x\) and \(y\) ?



A. \(4^{-x}\)

B. \(4^{-y}\)

C. \(2^{xy}\)

D. \(4^x\)

E. \(4^y\)

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[Reveal] Spoiler: OA

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Last edited by Carcass on 08 Feb 2019, 11:25, edited 1 time in total.
Edited by Carcass
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 20 May 2016, 18:00
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Explanation

Simplifying the fraction \(\frac{2^x^-^y}{2^x^+^y}\), yields \(\frac{2^x^-^y}{2^x^+^y}\)=\(2^-^2^y\). The only answer choice with 2 as the base is Choice C, \(2^x^y\), , which is clearly not the correct answer. Since all the other answer choices have 4 as the base, it is a good idea to rewrite 2–2y as an expression with 4 as the base, as follows.

\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).

Thus the correct answer is Choice B.
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 01 Jun 2016, 14:55
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sandy wrote:
Which of the following is equal to \(\frac{2^x^-^y}{2^x^+^y}\) for all integers x and y ?

A. \(4^-^x\)
B. \(4^-^y\)
C. \(2^x^y\)
D. \(4^x\)
E. \(4^y\)


Another approach is to use the INPUT-OUTPUT method.

Let x = 2 and y = 1


In this case, \(\frac{2^x^-^y}{2^x^+^y}\) = \(\frac{2^1}{2^3}\) = \(\frac{1}{4}\)
So, when the INPUT is x = 2 and y = 1, the OUTPUT is 1/4

The correct answer choice will be the one that has an OUTPUT of 1/4 when the INPUT is x = 2 and y = 1

A. \(4^-^x\) = 4^(-2) = 1/16. ELIMINATE A.
B. \(4^-^y\) = 4^(-1) = 1/4. KEEP B.
C. \(2^x^y\) = 2^(2) = 4. ELIMINATE C.
D. \(4^x\) = 4^(2) = 16. ELIMINATE D.
E. \(4^y\) = 4^(1) = 4. ELIMINATE E.

[Reveal] Spoiler:
B


RELATED RESOURCES (videos)
- Variables in the Answer Choices - https://www.greenlighttestprep.com/modu ... /video/936
- Tips for the Algebraic Approach - https://www.greenlighttestprep.com/modu ... /video/937
- Tips for the Input-Output Approach - https://www.greenlighttestprep.com/modu ... /video/938

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Brent
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 02 Aug 2018, 19:38
sandy wrote:
Explanation

Simplifying the fraction \(\frac{2^x^-^y}{2^x^+^y}\), yields \(\frac{2^x^-^y}{2^x^+^y}\)=\(2^-^2^y\). The only answer choice with 2 as the base is Choice C, \(2^x^y\), , which is clearly not the correct answer. Since all the other answer choices have 4 as the base, it is a good idea to rewrite 2–2y as an expression with 4 as the base, as follows.

\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).

Thus the correct answer is Choice B.


Can you please explain how you simplify that fraction?
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 26 Dec 2018, 05:56
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msawicka wrote:
sandy wrote:
Explanation

Simplifying the fraction \(\frac{2^x^-^y}{2^x^+^y}\), yields \(\frac{2^x^-^y}{2^x^+^y}\)=\(2^-^2^y\). The only answer choice with 2 as the base is Choice C, \(2^x^y\), , which is clearly not the correct answer. Since all the other answer choices have 4 as the base, it is a good idea to rewrite 2–2y as an expression with 4 as the base, as follows.

\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).

Thus the correct answer is Choice B.


Can you please explain how you simplify that fraction?


=\(\frac{2^x^-^y}{2^x^+^y}\)

=\(2^x^-^y^-^x^-^y\)

=\(2^-^2^y\)

now
\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 07 May 2019, 10:46
AE wrote:
msawicka wrote:
sandy wrote:
Explanation

Simplifying the fraction \(\frac{2^x^-^y}{2^x^+^y}\), yields \(\frac{2^x^-^y}{2^x^+^y}\)=\(2^-^2^y\). The only answer choice with 2 as the base is Choice C, \(2^x^y\), , which is clearly not the correct answer. Since all the other answer choices have 4 as the base, it is a good idea to rewrite 2–2y as an expression with 4 as the base, as follows.

\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).

Thus the correct answer is Choice B.


Can you please explain how you simplify that fraction?


=\(\frac{2^x^-^y}{2^x^+^y}\)

=\(2^x^-^y^-^x^-^y\)

=\(2^-^2^y\)

now
\(2^-^2^y= \frac{1}{2^2^y}= 4^-^y\).


What happens to the 2 on the bottom? I feel like I'm missing the step that allows us to put the exponents all on one line.
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Re: Which of the following is equal to..for all integers x and y [#permalink] New post 07 May 2019, 11:19
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It is a property of fraction and exponents.

The same base then you do have the two raised to the two exponents subtracted

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Re: Which of the following is equal to..for all integers x and y   [#permalink] 07 May 2019, 11:19
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