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If x and y are perfect squares, which of the following is

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If x and y are perfect squares, which of the following is [#permalink] New post 03 Oct 2011, 09:49
Can someone please help me to understand the explanation?
This is a question I came across in a GMAT preparation book (GMAT Prep Course by Jeff Kolby).

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

(is it supposed to mean "it might or it might not be a perfect square" or it is a definitive "it is not a perfect square"?)

(a) x^2
(b) xy
(c) 4x
(d) x+y
(e) x^5

The book solution is (d), because if you plug in x=4 and y=9, then x+y=13 which is not a perfect square. Ok, I cannot disagree with it, although if you plug in x=9 and y=16, then x+y=25 which is a perfect square. But can someone explain to me why (e) is not the right answer? x^5 also is not necessarily a square. 4^5 is a perfect square, but 3^5 or 5^5 are not. Help... Dumb book or dumb me?
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Re: Perfect squares [#permalink] New post 03 Oct 2011, 10:27
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Perfecr square means x^2 where x is an integer.
So, it is easy to see on each answer.

Note: not necessarily means either a perfect square or not.

Let assume x=a^2 and y=b^2, where a and b are integers.

(A)x^2=a^4, so, a perfect square
(B)xy=a^2b^2=(ab)^2, so, a perfect square
(C)4x=4a^2=(2a)^2, so, a perfect square
(D)x+y=a^2+b^2. Not necessarily a perfect square. See your own example.
(E)x^5=a^10=(a^5)^2, so, a perfect square.

All in all, the answer is (D)
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Re: Perfect squares [#permalink] New post 03 Oct 2011, 10:49
Bummer. That's me dumb indeed. I forgot x and y have to be squares too, that's why (e) is always a square. Thank you.
Re: Perfect squares   [#permalink] 03 Oct 2011, 10:49
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