Bunuel wrote:
Is the sum of x^y and y^x positive?
(1) xy > 0
(2) x + y > 0
Target question: Is the sum of x^y and y^x positive? Statement 1: xy > 0 There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 1. So, x^y + y^x = 1^1 + 1^1 = 2. In this case, the answer to the target question is
YES, the sum of x^y and y^x is positiveCase b: x = -1 and y = -1. So, x^y + y^x = (-1)^(-1) + (-1)^(-1) = -2. In this case, the answer to the target question is
NO, the sum of x^y and y^x is not positiveSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: x + y > 0There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = 1. So, x^y + y^x = 1^1 + 1^1 = 2. In this case, the answer to the target question is
YES, the sum of x^y and y^x is positiveCase b: x = 3 and y = -1. So, x^y + y^x = 3^(-1) + (-1)^3 = 1/3 + (-1) = -2/3. In this case, the answer to the target question is
NO, the sum of x^y and y^x is not positiveSince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1 (xy > 0) tells us that EITHER x and y are both positive OR x and y are both negative.
Statement 2 tells us that x + y is positive.
So, it must be the case that
x and y are both positive If x and y are both positive, then x^y is positive, and y^x is positive, which means
the sum of x^y and y^x is DEFINITELY positiveSince we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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