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07 Mar 2024, 00:56
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
59% (01:45) correct
41%
(01:36)
wrong
based on 22
sessions
History
Date
Time
Result
Not Attempted Yet
The value of \(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}}\) is
A. less than \(-10^6\)
B. greater than \(-10^6\) and less than \(-10^4\)
C. greater than \(-10^4\) and less than \(-10^2\)
D. greater than \(-10^2\) and less than \(0\)
E. greater than \(0\)
A. less than \(-10^6\)
B. greater than \(-10^6\) and less than \(-10^4\)
C. greater than \(-10^4\) and less than \(-10^2\)
D. greater than \(-10^2\) and less than \(0\)
E. greater than \(0\)
07 Mar 2024, 00:56
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Official Solution:
The value of \(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}}\) is
A. less than \(-10^6\)
B. greater than \(-10^6\) and less than \(-10^4\)
C. greater than \(-10^4\) and less than \(-10^2\)
D. greater than \(-10^2\) and less than \(0\)
E. greater than \(0\)
Expressions both in the numerator and denominator are quite ugly and do not factor nicely. However, looking at the answer options, we see that they are well spread, which should be a hint that we should approximate.
Observe that \(10^{20}\) is a significantly larger number than \(7^{5}\), which makes \(7^{5}\) negligible in comparison. Likewise, \(5^{20}\) is much, much larger than \(3^{5}\), making \(3^{5}\) also negligible. Therefore:
The above expression will be less than -1,000,000, which is \(-10^6\).
Answer: A
The value of \(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}}\) is
A. less than \(-10^6\)
B. greater than \(-10^6\) and less than \(-10^4\)
C. greater than \(-10^4\) and less than \(-10^2\)
D. greater than \(-10^2\) and less than \(0\)
E. greater than \(0\)
Expressions both in the numerator and denominator are quite ugly and do not factor nicely. However, looking at the answer options, we see that they are well spread, which should be a hint that we should approximate.
Observe that \(10^{20}\) is a significantly larger number than \(7^{5}\), which makes \(7^{5}\) negligible in comparison. Likewise, \(5^{20}\) is much, much larger than \(3^{5}\), making \(3^{5}\) also negligible. Therefore:
\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)
\(\approx\frac{-10^{20}}{5^{20}} = \)
\(=\frac{-2^{20}5^{20}}{5^{20}} = \)
\(=-2^{20} = \)
\(=-2^{10} 2^{10} \approx \)
\(\approx- 1,024*1,024\)
\(\approx\frac{-10^{20}}{5^{20}} = \)
\(=\frac{-2^{20}5^{20}}{5^{20}} = \)
\(=-2^{20} = \)
\(=-2^{10} 2^{10} \approx \)
\(\approx- 1,024*1,024\)
The above expression will be less than -1,000,000, which is \(-10^6\).
Answer: A
