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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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.



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

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

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

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

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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