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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.
_________________

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.

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