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GMAT Practice Question (One of Hundreds)
If x and y are both integers, which is larger, x^{x} or y^{y}?
x = y + 1
x^{y} > x and x is positive.
Correct Answer: C
For Statement (1), try plugging in negative values for x.
The problem deals with properties of exponents. Analyzing the different cases where x is positive and y is positive, for example, is the key to this problem.
Since x = y + 1, substitute for x in x^{x}.
x^{x} = (y + 1)^{(y + 1)}
Since x is one number larger than y, it may appear that x^{x} must be larger than y^{y}. However, consider the table below.
x
y
x^{x}
y^{y}
-1
-2
-1
1/4
-2
-3
1/4
-1/27
-3
-4
-1/27
1/256
When x = -1 and y = -2, x^{x} is smaller. However, when x = -2 and y = -3, y^{y} is smaller. Whether x^{x} or y^{y} is larger depends on the values of x and y.
Given the inequality from Statement (2),
x^{y} > x
Divide both sides by x.
x^{(y - 1)} > 1
First consider this inequality when y = 1. Then x^{(y - 1)} = x^{(1 - 1)} = 1. But this violates the inequality because it is not true that x^{(1 - 1)} > 1. Therefore, y may not be 1.
Next consider the case where y < 1. Then x^{(y - 1)} = x^{-k}, where -k is some negative number. And x^{-k} = 1 / x^{k}, which is less than 1 no matter the value of x; this violates the inequality, too, since x^{(y - 1)} is supposed to be greater than 1. For example, if y = -3 and x = 2, then x^{(y - 1)} = 2^{(-3 - 1)} = 1 / 2^{4} = 1/8, which is less than 1.
Since it cannot be that y = 1 or y < 1, the only option that remains is y > 1. From this conclusion and the information given in Statement (2), we conclude that x > 0 and y > 1. However, this is not enough information to determine whether x^{x} or y^{y} is larger. For example, it could be that x = 4 and y = 6; in this case, y^{y} would be larger. It could be that x = 7 and y = 3; in this case, x^{x} would be larger.
The conclusion reached in examining Statement (2) was that y > 1 and x > 0. Combine this with Statement (1), which says that x is one number larger than y. Thus, x^{x} will always be larger than y^{y}. For example, if y = 2, then x = 3; y^{y} = 2^{2} = 4 and x^{x} = 3^{3} = 27.
Statement (1) and (2) together are SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT yet Statements (1) and (2), when taken together, are SUFFICIENT, answer C is correct.