Problem Solving

If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2
(2) y > 2

Need help with the above problem.

With statement 1 you can plug in y^2 for x in the inequality in the question giving you this:

(y^2)^y < y^(y^2)

Simplified, you get:

y^2y < y^(y^2)

With each base as y, you can compare the exponents, or:

2y < y^2

If you continue to solve the expression is y < 2. Since we don't know the value of y, this is insufficient.

Statement 2 is clearly not sufficient because we know nothing about x.

However, together the simplified question from statement 1 is: "Is y < 2?" and statement 2 gives us that fact.

The answer is C

mohaprasad wrote:If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2
(2) y > 2

Need help with the above problem.

Statement 1: x = y²
Substituting y=x² into the question stem, we get:
(y²)^y < y^y²
y^(2y) < y^y².

Question stem, rephrased:
Is y^(2y) < y^y²?

Test one case that also satisfies statement 2.
Case 1: y=3
Plugging y=3 into the y^(2y) < y^y², we get:
3� < 3�.
YES.

Test one case that doesn't also satisfy statement 2.
Case 2: y=1
Plugging y=1 into y^(2y) < y^y², we get:
1² < 1¹
NO.

Since the answer is YES in Case 1 but NO in case 2, insufficient.

Statement 2:
Case 1 also satisfies statement 2.
In Case 1, the answer to the question stem is YES.

Case 3: y=4, x=-1
Plugging x=-1 and y=4 into x^y < y^x, we get:
(-1)� < 4^(-1)
1 < 1/4.
NO.

Since the answer to the question stem is YES in Case 1 but NO in Case 3, INSUFFICIENT.

Statements combined:
Case 1 satisfies both statements.
In Case 1, the answer to the question stem is YES.

Test an extreme case that satisfies both statements.
Case 4: y=10
Plugging y=10 into y^(2y) < y^y², we get:
10²� < 10¹��.
YES.

Case 4 illustrates that -- when both statements are satisfied -- the lefthand side will always be less than the righthand side.
SUFFICIENT.

The correct answer is C.