OfficialSolution:
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 negativeeven 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
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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
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 negativeeven 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
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Statistics : Posted by Bunuel • on 21 Jun 2024, 01:31 • Replies 3 • Views 426
