BrentGMATPrepNow wrote:
Is \(xy < 0\)?
(1) \(\frac{3^x}{3^y} = (3^x)(3^y)\)
(2) \((x + y)^2 = y^2\)
Target question: Is \(xy < 0\)? Statement 1: \(\frac{3^x}{3^y} = (3^x)(3^y)\) Apply the Quotient Law to the left side and the Product Law to the right side: \(3^{x-y} = 3^{x+y}\)
Since the bases are equal (and the bases don't equal -1, 0 or 1), the exponents must be equal: \(x-y = x+y\)
Add \(y\) to both sides: \(x = x+2y\)
Subtract \(x\) from both sides: \(0 = 2y\), which means \(y=0\)
So, regardless of the value of \(x\), we know that the product \(xy\) must equal \(0\), which means the answer to the target question is
NO, xy is not less than zeroSince we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: \((x + y)^2 = y^2\) Expand and simplify the left side to get: \(x^2 + 2xy + y^2 = y^2\)
Subtract \(y^2\) from both sides: \(x^2 + 2xy = 0\)
Factor the left side: \(x(x + 2y) = 0\)
So, EITHER \(x = 0\), OR \(x+2y=0\)
Let's examine each case:
Case a: If \(x = 0\), then the product \(xy\) must equal \(0\), which means the answer to the target question is
NO, xy is not less than zeroCase b: If \(x+2y=0\), then it could be the case that \(x = 2\) and \(y = -1\), in which case \(xy = (2)(-1) = -2\), which means the answer to the target question is
YES, xy is less than zeroSince we can’t answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
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