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705-805 Level|   Arithmetic|   Exponents|      
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Bunuel
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The value of (7^5 - 10^20)(5^20 - 3^5) is 07 Mar 2024, 00:58
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A
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85% (hard)

Question Stats:

52% (02:07) correct 48% (02:29) wrong based on 337 sessions

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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 \(0\)­
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A
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Bunuel
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The value of (7^5 - 10^20)(5^20 - 3^5) is 15 Mar 2024, 14:00
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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:

\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)

\(\approx\frac{-10^{20}}{5^{20}} = \)

\(=\frac{-2^{20}5^{20}}{5^{20}} = \)

\(=-2^{20} = \)

\(=-2^{10} 2^{10} \approx \)

\(\approx- 1,024*1,024\)

The above expression will be less than -1,000,000, which is \(-10^6\).


Answer: A­
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The value of (7^5 - 10^20)(5^20 - 3^5) is 07 Mar 2024, 01:10
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Bunuel
­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 \(0\)­
­We can approximate the values 

\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \frac{- 10^{20}}{5^{20}}\)

\(\approx - 2^{20} \approx - 2^{10} * 2^{10} \)

\(\approx - 1024 * 1024 \)

\(\approx -10^6 *\) Some Value

Hence the final value will be less than \(-10^6\)

Option A­
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The value of (7^5 - 10^20)(5^20 - 3^5) is 28 May 2024, 12:02
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Bunuel
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:




\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)

\(\approx\frac{-10^{20}}{5^{20}} = \)

\(=\frac{-2^{20}5^{20}}{5^{20}} = \)

\(=-2^{20} = \)

\(=-2^{10} 2^{10} \approx \)

\(\approx- 1,024*1,024\)

The above expression will be less than -1,000,000, which is \(-10^6\).


Answer: A­
­I thought it would be positive because raising -2 to an even power (i.e. 10) would result in positive 1024­. Could you clarify that please?
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The value of (7^5 - 10^20)(5^20 - 3^5) is 29 May 2024, 00:01
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atova01
Bunuel
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:






\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)

\(\approx\frac{-10^{20}}{5^{20}} = \)

\(=\frac{-2^{20}5^{20}}{5^{20}} = \)

\(=-2^{20} = \)

\(=-2^{10} 2^{10} \approx \)

\(\approx- 1,024*1,024\)

The above expression will be less than -1,000,000, which is \(-10^6\).


Answer: A­
­I thought it would be positive because raising -2 to an even power (i.e. 10) would result in positive 1024­. Could you clarify that please?
If it were \((­-2)^{10}\) youd be correct becasue this means -2 raised to the power of 10, which would be positive. However, \(-2^{10}\) essentially means \(-1*2^{10}\) , so it's negative.­
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The value of (7^5 - 10^20)(5^20 - 3^5) is 26 Jun 2025, 10:28
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Bunuel why are we ignoring 7 and 3 in solution
Bunuel
atova01
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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:






\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)

\(\approx\frac{-10^{20}}{5^{20}} = \)

\(=\frac{-2^{20}5^{20}}{5^{20}} = \)

\(=-2^{20} = \)

\(=-2^{10} 2^{10} \approx \)

\(\approx- 1,024*1,024\)

The above expression will be less than -1,000,000, which is \(-10^6\).


Answer: A­
­I thought it would be positive because raising -2 to an even power (i.e. 10) would result in positive 1024­. Could you clarify that please?
If it were \((­-2)^{10}\) youd be correct becasue this means -2 raised to the power of 10, which would be positive. However, \(-2^{10}\) essentially means \(-1*2^{10}\) , so it's negative.­
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Bunuel
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The value of (7^5 - 10^20)(5^20 - 3^5) is 26 Jun 2025, 10:33
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shubhim20
Bunuel why are we ignoring 7 and 3 in solution
Bunuel
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:


\(\frac{7^{5} - 10^{20}}{5^{20} - 3^{5}} \approx \)

\(\approx\frac{-10^{20}}{5^{20}} = \)

\(=\frac{-2^{20}5^{20}}{5^{20}} = \)

\(=-2^{20} = \)

\(=-2^{10} 2^{10} \approx \)

\(\approx- 1,024*1,024\)

The above expression will be less than -1,000,000, which is \(-10^6\).

Answer: A­

I think it is explained in the solution, no? Check the highlighted part.
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