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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
