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05 Jun 2024, 02:52
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If \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}\), and \(y^x=-\frac{1}{64}\), what is the value of \(xy\)?
A. 12
B. 18
C. 24
D. 32
E. 64
A. 12
B. 18
C. 24
D. 32
E. 64
05 Jun 2024, 02:52
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Official Solution:
If \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}\), and \(y^x=-\frac{1}{64}\), what is the value of \(xy\)?
A. 12
B. 18
C. 24
D. 32
E. 64
Since both \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}\) implies that \(x^y=(-9)^{-2}\) or \(x^y=(-3)^{-4}\), 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.
For \(y^x=-\frac{1}{64}\), as the result is negative, \(x\) must be a negative odd number. Therefore, \(y^x=-\frac{1}{64}\) implies that \(y^x=(-4)^{-3}\) or \(y^x=(-64)^{-1}\), and \(xy\) could be 12 or 64.
Only one pair of negative integers, \(x\) and \(y\), satisfies both conditions: \(x=-3\) and \(y=-4\). Thus, \(xy=12\).
Answer: A
If \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}\), and \(y^x=-\frac{1}{64}\), what is the value of \(xy\)?
A. 12
B. 18
C. 24
D. 32
E. 64
Since both \(x\) and \(y\) are negative integers, \(x^y=\frac{1}{81}\) implies that \(x^y=(-9)^{-2}\) or \(x^y=(-3)^{-4}\), 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.
For \(y^x=-\frac{1}{64}\), as the result is negative, \(x\) must be a negative odd number. Therefore, \(y^x=-\frac{1}{64}\) implies that \(y^x=(-4)^{-3}\) or \(y^x=(-64)^{-1}\), and \(xy\) could be 12 or 64.
Only one pair of negative integers, \(x\) and \(y\), satisfies both conditions: \(x=-3\) and \(y=-4\). Thus, \(xy=12\).
Answer: A
27 Jul 2024, 11:37
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This question has been wrongly tagged as easy in the GMAT FOCUS Test. Due to which we are getting inaccurate test score Bunuel
