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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 ?

Re: If X & Y are perfect squares, then which one of the [#permalink]
05 Jun 2012, 02:53

X = m^2; m = integer. Now X^5 = m^10 which is a perfect square (root being m^5).
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Re: If X & Y are perfect squares, then which one of the [#permalink]
05 Jun 2012, 02:57

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
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Re: If x and y are perfect squares, then which one of the [#permalink]
22 Apr 2013, 23:31

sdas wrote:

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

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.

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

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.
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