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16 Sep 2014, 01:26
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If \(x\) and \(y\) are negative integers, what is the value of \(xy\)?
(1) \(x^y=\frac{1}{81}\)
(2) \(y^x=-\frac{1}{64}\)
(1) \(x^y=\frac{1}{81}\)
(2) \(y^x=-\frac{1}{64}\)
16 Sep 2014, 01:26
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Official Solution:
If \(x\) and \(y\) are negative integers, what is the value of \(xy\)?
(1) \(x^y=\frac{1}{81}\).
Since both \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}\), so \(xy\) could be 18 or 12. Note that for a negative integer (\(x\)) raised to a negative integer power (\(y\)) to yield a positive number (\(\frac{1}{81}\)), the power must be a negative even number. Not sufficient.
(2) \(y^x=-\frac{1}{64}\).
As the result is negative, \(x\) must be a negative odd number. Therefore, \(y^x=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}\), and \(xy\) could be 12 or 64. Not sufficient.
(1)+(2) Only one pair of negative integers, \(x\) and \(y\), satisfies both statements: \(x=-3\) and \(y=-4\). Thus, \(xy=12\). Sufficient.
Answer: C
If \(x\) and \(y\) are negative integers, what is the value of \(xy\)?
(1) \(x^y=\frac{1}{81}\).
Since both \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}\), so \(xy\) could be 18 or 12. Note that for a negative integer (\(x\)) raised to a negative integer power (\(y\)) to yield a positive number (\(\frac{1}{81}\)), the power must be a negative even number. Not sufficient.
(2) \(y^x=-\frac{1}{64}\).
As the result is negative, \(x\) must be a negative odd number. Therefore, \(y^x=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}\), and \(xy\) could be 12 or 64. Not sufficient.
(1)+(2) Only one pair of negative integers, \(x\) and \(y\), satisfies both statements: \(x=-3\) and \(y=-4\). Thus, \(xy=12\). Sufficient.
Answer: C
General Discussion
13 Jul 2017, 07:54
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Bunuel
Very good question.
In statement 2, I forget the SPECIAL CASE in which x = -1.
