If x and y are perfect squares, then which one of the

Question

If x and y are perfect squares, then which one of the following is not necessarily a perfect square?

A. x^2
B. xy
C. 4x
D. x+y
E. x^5

Source:   Nova GMAT  
Difficulty Level:   550

Seems like a straight forward problem, however the answer choices are a little confusing .

both D& E seem to be contendors for the right answer though the OA is D

D-> Because the sum of two perfect square needn't necessarily be a perfect square e.g 4+9=13, not a perfect square, however 9 & 16 = 25 which is a perfect square)
E-> Because a perfect square to the power 5 (an ODD number) cannot be a perfect square right ?

Please explain.

Thanks,
Shreya

AbhiJ · Answer

X = m^2; m = integer.
Now X^5 = m^10 which is a perfect square (root being m^5).

pike · Answer

x = a^2
y = b^2

a) (a^2)^2, ok
b) a^2 * b^2 = (ab)^2, ok
c) 4*a^2 = 2^2*a^2 = (2a)^2, ok
d) a^2 + b^2, clearly not ok for all a/b, e.g a = 8, b = 2, 64 + 4 = 68 which is not a perfect square.
e) (a^2)^5 = (a^5)^2, ok

sdas · Answer

Option B is not clear to me....
X=a^2
Y=b^2

Therefore X*Y = (ab)^2
But, if x=4 and y=9 , then xy=25
However, if x= 4 and y = 25, then xy = 100 which is not perfect square

So, how can we rule out B?

Please explain

mdbharadwaj · Answer

If x=4 and y=9 xy =36 which is a perfect square. and xy=100 is a perfect square. its the square of 10.
for that matter , product of any two perfect squares is a perfect square. as you pointed out (ab)^2. so square root of the product is ab an integer.

4*9 or 9*25 or 16*25

mau5 · Answer

XY = 36 = 6^2

XY for x=4 and Y = 25 is 100 = 10^2, which is a perfect square.

Bunuel · Answer

If x=y=1^2=1, then each option but D is a perfect square, therefore D is NOT  necessarily  a perfect square.

Answer: D.

P.S. Notice that x+y could be a perfect square for example if x=3^2=9 and y=4^2=16 --> x+y=25=5^2.

sdas · Answer

Darn....I forgot perfect square of 10.

shwetakoshija · Answer

Since x and y are perfect squares, we can write:
x=n^2 and y=m^2, for some integers m and n.

Let's evaluate the options now:
1. x^2 = (n^2)^2. Thus, x^2 is a perfect square (the square of n^2).
2. xy = (n^2)(m^2)= (nm)^2. Thus, xy is a perfect square (the square of xy)
3. 4x = 4(n^2)= (2n)^2. Thus, 4x is a perfect square (the square of 2n)
4. x+y= (n^2)+(m^2). This is NOT necessarily a perfect square.

Hence, we have our answer. It is D.

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