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^[(xy)^2]}?
A. 2
B. 4
C. 8
D. 16
E. 32
Any help/advice would be much appreciated!
Practice Quant problem
This topic has expert replies

 Newbie  Next Rank: 10 Posts
 Posts: 5
 Joined: 17 Mar 2019
GMAT/MBA Expert
 [email protected]
 GMAT Instructor
 Posts: 3008
 Joined: 22 Aug 2016
 Location: Grand Central / New York
 Thanked: 470 times
 Followed by:32 members
We have {2^[(x+y)^2]}/{2^[(xy)^2]}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^[(xy)^2]}?
A. 2
B. 4
C. 8
D. 16
E. 32
Any help/advice would be much appreciated!
{2^[(x+y)^2]}/{2^[(xy)^2]} = 2^[{(x+y)^2}  {(xy)^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^[(xy)^2]} = {2^4}/{2^0} = 16/1 = 16  Correct answer is D.
Hope this helps!
Jay
_________________
Manhattan Review GMAT Prep
Locations: GRE Manhattan  ACT Prep Courses San Diego  IELTS Prep Courses Seattle  Dallas IELTS Tutoring  and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
GMAT/MBA Expert
 Ian Stewart
 GMAT Instructor
 Posts: 2612
 Joined: 02 Jun 2008
 Location: Toronto
 Thanked: 1090 times
 Followed by:355 members
 GMAT Score:780
2^a / 2^b is equal to 2^(ab).
So 2^(x+y)^2 / 2^(xy)^2 is equal to 2^[ (x+y)^2  (xy)^2 ]
We could expand both (x+y)^2 and (xy)^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  (xy)^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.
So 2^(x+y)^2 / 2^(xy)^2 is equal to 2^[ (x+y)^2  (xy)^2 ]
We could expand both (x+y)^2 and (xy)^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  (xy)^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.
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

 Junior  Next Rank: 30 Posts
 Posts: 17
 Joined: 14 Aug 2020
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(x2)) = 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.
( x + y )^2 = x^2 + 2xy + y^2
( x  y )^2 = x^2  2xy + y^2
2^x / 2^x  2 = 2^(x(x2)) = 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.