Practice Quant problem

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myspecialtie: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Sun Mar 17, 2019 10:38 am

by myspecialtie » Mon Mar 25, 2019 11:17 am

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!

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Source: Beat The GMAT — Quantitative Reasoning | Mon Mar 25, 2019 11:17 am

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Mon Apr 22, 2019 11:00 pm

myspecialtie wrote: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: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Mon Jun 03, 2019 6:38 am

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.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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PhillipZhang87: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Fri Aug 14, 2020 9:44 am

Re: Practice Quant problem

by PhillipZhang87 » Fri Sep 18, 2020 7:03 am

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.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8139; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: Practice Quant problem

by Scott@TargetTestPrep » Fri Jun 17, 2022 11:37 am

myspecialtie wrote: ↑
Mon Mar 25, 2019 11:17 am
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

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