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16 Sep 2014, 01:31
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If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?
I. \(x=1\)
II. \(x=1\) and \(y=0\)
III. \(x=1\) or \(y=0\)
A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)
I. \(x=1\)
II. \(x=1\) and \(y=0\)
III. \(x=1\) or \(y=0\)
A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)
16 Sep 2014, 01:31
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Official Solution:
If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?
I. \(x=1\)
II. \(x=1\) and \(y=0\)
III. \(x=1\) or \(y=0\)
A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)
\(x^y=1\) implies we have one of the following three cases:
1. The base is 1, because \(1^y=1\), for all \(y\);
2. The exponent is 0, because \(x^0=1\), for any nonzero \(x\);
3. The base is -1 and the exponent is even, because \((-1)^x=1\), for any even \(x\)
Notice that if we have case 3, so if \(x=-1\) and \(y\) is any even number, then \((-1)^{even}=1\), and in this case none of the options must be true.
Answer: E
If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?
I. \(x=1\)
II. \(x=1\) and \(y=0\)
III. \(x=1\) or \(y=0\)
A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)
\(x^y=1\) implies we have one of the following three cases:
1. The base is 1, because \(1^y=1\), for all \(y\);
2. The exponent is 0, because \(x^0=1\), for any nonzero \(x\);
3. The base is -1 and the exponent is even, because \((-1)^x=1\), for any even \(x\)
Notice that if we have case 3, so if \(x=-1\) and \(y\) is any even number, then \((-1)^{even}=1\), and in this case none of the options must be true.
Answer: E
09 Jul 2020, 11:25
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I think this is a high-quality question and I agree with explanation. An excellent question
