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Retired Moderator Joined: 07 Jun 2014
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Which of the following is equal to..for all integers x and y [#permalink]
Expert's post 00:00

Question Stats: 78% (00:56) correct 21% (01:04) wrong based on 121 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. $$4^y$$

Practice Questions
Question: 10
Page: 83
[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 Retired Moderator Joined: 07 Jun 2014
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Re: Which of the following is equal to..for all integers x and y [#permalink]
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Expert's post
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]
Expert's post
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

Cheers,
Brent
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Re: Which of the following is equal to..for all integers x and y [#permalink]
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? Director Joined: 09 Nov 2018
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Re: Which of the following is equal to..for all integers x and y [#permalink]
1
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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]
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]
Expert's post
It is a property of fraction and exponents.

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

Attachment: GMAT Club Math Book v3 - Jan-2-2013.pdf [2.83 MiB]

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