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Subhashghosh is correct, as (-1)^-1 stays negative, while (-2)^-2 becomes positive.

-1 < 1/4

This is an odd problem, though. I agree that it should mention that x and y aren't 0. It also shouldn't ask which option is greater, as this rules out in advance the possibility that the two expressions are equal. A real GMAT question would typically ask if one or the other was greater, not *which* expression is greater.
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Dmitry Farber | Manhattan GMAT Instructor | New York

The option here should say x and y != 0 as GMAT does not test 0^0.

(2)

xy > x and x is positive

=> y > 0

But x can be > y or y can be > x

Not Sufficient

(1) + (2)

x^x > y^y

Answer - C

doesnt this statement: "xy > x and x is positive" actually mean

==> y>1

if this is the case i am guessing statement 2 will be sufficient to solve the question. Am i making any mistake here?

If x and y are both integers, which is larger, x^x or y^y?

(1) x = y + 1 --> if y is positive integer then x^x=(y+1)^{y+1}>y^y but if y=-2 then x=-1 and x^x=-1<\frac{1}{4}=y^y

(2) x^y > x and x is positive --> since x is positive then x^{y-1}>1 --> since x and y are integers then y>1. If x=1 and y=2 then x^x<y^y but if x=3 and y=2 then x^x>y^y. Not sufficient.

(1)+(2) From (2) y>1, so it's a positive integer then from (1) x^x=(y+1)^{y+1}>y^y. Sufficient.