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26 May 2017, 04:48
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Which of the following ranges contains the approximation of \(\frac{5^{10}+5^{20}}{10^{10}+10^{20}}\)?
A. ~0.000001
B. 0.000001~0.00001
C. 0.00001~0.0001
D. 0.0001~0.001
E. 0.001~
A. ~0.000001
B. 0.000001~0.00001
C. 0.00001~0.0001
D. 0.0001~0.001
E. 0.001~
26 May 2017, 04:49
Kudos
Bookmarks
Official Solution:
Which of the following ranges contains the approximation of \(\frac{5^{10}+5^{20}}{10^{10}+10^{20}}\)?
A. ~0.000001
B. 0.000001~0.00001
C. 0.00001~0.0001
D. 0.0001~0.001
E. 0.001~
\(\frac{5^{10}+5^{20}}{10^{10}+10^{20}} = \frac{5^{10}+5^{10}5^{10}}{10^{10}+10^{10}10^{10}}= \frac{5^{10}(1+5^{10})}{10^{10}(1+10^{10})}\), and since there is no point in adding 1 to the numerator and the denominator, you get
\(\frac{5^{10}(1 + 5^{10})}{10^{10}(1 + 10^{10})} ≒ \frac{5^{10}(5^{10})}{10^{10}(10^{10})}\) = \(\frac{5^{20}}{10^{20}} =\) \((\frac{5}{10})^{20} =\) \((\frac{1}{2})^{20}\) = \(\frac{1}{2^{20}}\)
However, from \(2^{10} = 1,024 > 1,000 = 10^{3}\), if you square both sides, you get \((2^{10})^{2} > (10^{3})^{2},\) or \(2^{20} > 10^{6}\), then \(\frac{1}{2^{20}} < \frac{1}{10^{6}} = 0.000001\). Therefore, the answer is A
Answer: A
Which of the following ranges contains the approximation of \(\frac{5^{10}+5^{20}}{10^{10}+10^{20}}\)?
A. ~0.000001
B. 0.000001~0.00001
C. 0.00001~0.0001
D. 0.0001~0.001
E. 0.001~
\(\frac{5^{10}+5^{20}}{10^{10}+10^{20}} = \frac{5^{10}+5^{10}5^{10}}{10^{10}+10^{10}10^{10}}= \frac{5^{10}(1+5^{10})}{10^{10}(1+10^{10})}\), and since there is no point in adding 1 to the numerator and the denominator, you get
\(\frac{5^{10}(1 + 5^{10})}{10^{10}(1 + 10^{10})} ≒ \frac{5^{10}(5^{10})}{10^{10}(10^{10})}\) = \(\frac{5^{20}}{10^{20}} =\) \((\frac{5}{10})^{20} =\) \((\frac{1}{2})^{20}\) = \(\frac{1}{2^{20}}\)
However, from \(2^{10} = 1,024 > 1,000 = 10^{3}\), if you square both sides, you get \((2^{10})^{2} > (10^{3})^{2},\) or \(2^{20} > 10^{6}\), then \(\frac{1}{2^{20}} < \frac{1}{10^{6}} = 0.000001\). Therefore, the answer is A
Answer: A
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