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16 Sep 2014, 01:24
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If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\), what is the value of \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}\)?
A. \(\frac{14}{5}\)
B. 5
C. \(\frac{28}{5}\)
D. 13
E. 14
A. \(\frac{14}{5}\)
B. 5
C. \(\frac{28}{5}\)
D. 13
E. 14
16 Sep 2014, 01:24
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Official Solution:
If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\), what is the value of \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}\)?
A. \(\frac{14}{5}\)
B. 5
C. \(\frac{28}{5}\)
D. 13
E. 14
The first thing one should notice here is that \(x\) and \(y\) must be some irrational numbers. This is because 4,900 has primes other than 5 in its prime factorization, and 25 doesn't have 2 as a prime at all. Thus, we should manipulate the given expressions rather than solve for \(x\) and \(y\) directly.
\(5^{10x}=4,900\)
\((5^{5x})^2=70^2\)
\(5^{5x}=70\)
Now, \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}=5^{(5x-5)}*4^{\sqrt{y}} = 5^{5x}*5^{-5}*(2^{\sqrt{y}})^2\)
Since \(5^{5x}=70\), then \(5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2=70*5^{-5}*5^4=70*5^{-1}=\frac{70}{5}=14\)
Answer: E
If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\), what is the value of \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}\)?
A. \(\frac{14}{5}\)
B. 5
C. \(\frac{28}{5}\)
D. 13
E. 14
The first thing one should notice here is that \(x\) and \(y\) must be some irrational numbers. This is because 4,900 has primes other than 5 in its prime factorization, and 25 doesn't have 2 as a prime at all. Thus, we should manipulate the given expressions rather than solve for \(x\) and \(y\) directly.
\(5^{10x}=4,900\)
\((5^{5x})^2=70^2\)
\(5^{5x}=70\)
Now, \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}=5^{(5x-5)}*4^{\sqrt{y}} = 5^{5x}*5^{-5}*(2^{\sqrt{y}})^2\)
Since \(5^{5x}=70\), then \(5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2=70*5^{-5}*5^4=70*5^{-1}=\frac{70}{5}=14\)
Answer: E
General Discussion
I think this is a high-quality question and I agree with explanation.
