Practice Quant problem

[myspecialtie]

I'm having difficulty figuring out the solution to this problem from one of the GMAT practice exams:

If xy=1, what is the value of {2^[(x+y)^2]}/{2^[(x-y)^2]}?

A. 2
B. 4
C. 8
D. 16
E. 32

Any help/advice would be much appreciated!

[Jay@ManhattanReview]

I'm having difficulty figuring out the solution to this problem from one of the GMAT practice exams:

If xy=1, what is the value of {2^[(x+y)^2]}/{2^[(x-y)^2]}?

A. 2
B. 4
C. 8
D. 16
E. 32

Any help/advice would be much appreciated!

We have {2^[(x+y)^2]}/{2^[(x-y)^2]}

{2^[(x+y)^2]}/{2^[(x-y)^2]} = 2^[{(x+y)^2} - {(x-y)^2}] = 2^[{x^2 + y^2 + 2xy} - {x^2 + y^2 - 2xy}] = 2^(4xy) = 2^(4*1) = 2^4 = 16

The correct answer: D

Alternate approach:

Since we can choose any value for x or y given that xy = 1, say x = y = 1

Thus, (x + y)^2 = (1 + 1)^2 = 2^2 = 4 and (x - y)^2 = (1 - 1)^2 = 0^2 = 0

{2^[(x+y)^2]}/{2^[(x-y)^2]} = {2^4}/{2^0} = 16/1 = 16 -- Correct answer is D.

Hope this helps!

-Jay
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[Ian Stewart]

2^a / 2^b is equal to 2^(a-b).

So 2^(x+y)^2 / 2^(x-y)^2 is equal to 2^[ (x+y)^2 - (x-y)^2 ]

We could expand both (x+y)^2 and (x-y)^2 and subtract, but it's faster just to use the difference of squares factorization immediately, since we're subtracting one square from another in the exponent. So our exponent is equal to

(x+y)^2 - (x-y)^2 = (x + y + x - y) ( x + y - x + y) = (2x)(2y) = 4xy

and since xy = 1, our exponent is equal to 4, so the overall value of the expression is 2^4 = 16.

[PhillipZhang87]

For this question we have to know the following:
( x + y )^2 = x^2 + 2xy + y^2
( x - y )^2 = x^2 - 2xy + y^2
2^x / 2^x - 2 = 2^(x-(x-2)) = 2^2
Moving to the problem
2^(x^2 + 2xy + y^2) / 2^(x^2 - 2xy + y^2) = 2^4xy
We know xy = 1
So, 2^4(1) = 2^4 = 16.
Correct answer is D.

[Scott@TargetTestPrep]

I'm having difficulty figuring out the solution to this problem from one of the GMAT practice exams:

If xy=1, what is the value of {2^[(x+y)^2]}/{2^[(x-y)^2]}?

A. 2
B. 4
C. 8
D. 16
E. 32

Any help/advice would be much appreciated!

We can simplify the given expression:

[2^(x+y)^2]/[2^(x-y)^2]

Expanding the exponents in both the numerator and the denominator, we have:

[2^(x^2+y^2+2xy)]/[2^(x^2+y^2-2xy]

We subtract the denominator’s exponent from the numerator’s exponent:

2^(x^2 + y^2 + 2xy - x^2 - y^2 + 2xy)

2^(2xy + 2xy) = 2^(4xy)

Since xy = 1, 2^4xy = 2^4 = 16.

Answer: D