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705-805 Level|   Arithmetic|   Exponents|      
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 29 Dec 2023, 08:27
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

75% (hard)

Question Stats:

65% (02:18) correct 35% (02:35) wrong based on 1101 sessions

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The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

(A) less than 0.00005
(B) greater than 0.00005 and less than 0.05
(C) greater than 0.05 and less than 50
(D) greater than 50 and less than 50,000
(E) greater than 50,000
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E
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 03 Jan 2024, 09:45
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Asked: The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

\(\frac{8^4 + 8^{16}}{4^8 + 16^8} = \frac{2^{12} + 2^{48} }{ 2^{16} + 2^{32}} = \frac{2^{12} (1 + 2^{36})}{ 2^{16} (1 + 2^{16})} = \frac{1 + 2^{36}}{ 2^4 (1 + 2^{16})} = \frac{2^{36}}{2^4*2^{16}} = 2^{16} = 64*1024 > 50,000 \)

IMO E
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 11 Jan 2024, 02:28
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Sazimordecai
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The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

(A) less than 0.00005
(B) greater than 0.00005 and less than 0.05
(C) greater than 0.05 and less than 50
(D) greater than 50 and less than 50,000
(E) greater than 50,000


\(\frac{8^4 + 8^{16}}{4^8 + 16^8}\)

\(\frac{8^4 + 8^{16}}{4^8 + 4^{16}}\)

\(8^{16} >> 8^4\), hence \(8^4 + 8^{16} \approx 8^{16}\)

\(4^{16} >> 4^8\), hence \(4^8 + 4^{16} \approx 4^{16}\)

\(\approx \frac{8^{16}}{4^{16}}\)

\(\approx \frac{(2*4)^{16}}{4^{16}}\)

\(\approx \frac{2^{16}*4^{16}}{4^{16}}\)

\(\approx 2^{16} = 2^{10}*2^{6} \approx 1024 * 64 \approx 64000 \)

Option E


If I may ask : why are you saying 8^4 + 8^16 is equivalent to 8^16

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That's because 8^16 is significantly larger than 8^4, making it essentially negligible in the sum 8^16 + 8^4. Here is why:

    \(8^{16} + 8^4 = 8^4(8^{12} + 1)\)

8^12 + 1 is essentially 8^12. Hence, \(8^4(8^{12} + 1)\approx 8^4*8^{12}=8^{16}\).

Similarly, 16^8 is so much larger than 4^8 that it's also negligible in the sum 4^8 + 16^8. Therefore:

    \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\approx\)

    \(\approx \frac{8^{16}}{16^8}=\)

    \(=\frac{2^{48}}{2^{32}}=\)

    \(=2^{16}=\)

    \(=2^{10}*2^6=\)

    \(\approx 1,000*64 > 50,000\).

Answer: E.

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Hope it helps.
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 29 Dec 2023, 09:03
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The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

(A) less than 0.00005
(B) greater than 0.00005 and less than 0.05
(C) greater than 0.05 and less than 50
(D) greater than 50 and less than 50,000
(E) greater than 50,000
Show more


\(\frac{8^4 + 8^{16}}{4^8 + 16^8}\)

\(\frac{8^4 + 8^{16}}{4^8 + 4^{16}}\)

\(8^{16} >> 8^4\), hence \(8^4 + 8^{16} \approx 8^{16}\)

\(4^{16} >> 4^8\), hence \(4^8 + 4^{16} \approx 4^{16}\)

\(\approx \frac{8^{16}}{4^{16}}\)

\(\approx \frac{(2*4)^{16}}{4^{16}}\)

\(\approx \frac{2^{16}*4^{16}}{4^{16}}\)

\(\approx 2^{16} = 2^{10}*2^{6} \approx 1024 * 64 \approx 64000 \)

Option E
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 11 Jan 2024, 01:49
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gmatophobia
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The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

(A) less than 0.00005
(B) greater than 0.00005 and less than 0.05
(C) greater than 0.05 and less than 50
(D) greater than 50 and less than 50,000
(E) greater than 50,000


\(\frac{8^4 + 8^{16}}{4^8 + 16^8}\)

\(\frac{8^4 + 8^{16}}{4^8 + 4^{16}}\)

\(8^{16} >> 8^4\), hence \(8^4 + 8^{16} \approx 8^{16}\)

\(4^{16} >> 4^8\), hence \(4^8 + 4^{16} \approx 4^{16}\)

\(\approx \frac{8^{16}}{4^{16}}\)

\(\approx \frac{(2*4)^{16}}{4^{16}}\)

\(\approx \frac{2^{16}*4^{16}}{4^{16}}\)

\(\approx 2^{16} = 2^{10}*2^{6} \approx 1024 * 64 \approx 64000 \)

Option E
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If I may ask : why are you saying 8^4 + 8^16 is equivalent to 8^16

Posted from my mobile device
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 11 Jan 2024, 09:14
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2^12 (1+2^36 )/2^16 )(1+2^16 )=(1/2^4)(2^36 )/2^16))=2^16

2^6=64, 2^10=1024

2^16 = 64000

E)
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 24 Jun 2024, 01:07
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The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

(A) less than 0.00005
(B) greater than 0.00005 and less than 0.05
(C) greater than 0.05 and less than 50
(D) greater than 50 and less than 50,000
(E) greater than 50,000
Show more
­Here is a video on how exponents are placed on the number line: https://youtu.be/0rpppnnJNRs
It shows you the relative value of exponents. And here is a video on Estimation: https://youtu.be/4Wy7BrQrjkM

\(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) 

\(\frac{2^{12} + 2^{48}}{2^{16} + 2^{32}}\)  (Bringing everything in terms of 2)

The moment you see this expression and look at the options, you should be able to say that 2^12 is extremely small compared to 2^48 so it can be ignored. Also 2^16 is extremely small compared to 2^32 and hence should be ignored too. 

\(\frac{2^{48}}{2^{32}} = 2^{16} = 2^{10}*2^6 = 1000 * 64 = 64000\) (approximately)
 ­
Answer (E)
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 02 Jul 2024, 00:48
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­Watch this solution to see how you should be converting this question into a time saver question, saving you at least one minute in the test environment.

 ­­
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 12 Jul 2024, 11:27
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Asked: The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

\(\frac{8^4 + 8^{16}}{4^8 + 16^8} = \frac{2^{12} + 2^{48} }{ 2^{16} + 2^{32}} = \frac{2^{12} (1 + 2^{36})}{ 2^{16} (1 + 2^{16})} = \frac{1 + 2^{36}}{ 2^4 (1 + 2^{16})} = \frac{2^{36}}{2^4*2^{16}} = 2^{16} = 64*1024 > 50,000 \)

IMO E
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­I udnerstand up until this part - what happened to the "1" in the first row?

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The value of (8^4 + 8^16)/(4^8 + 16^8) is 12 Jul 2024, 11:46
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Danou

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Asked: The value of \(\frac{8^4 + 8^{16}}{4^8 + 16^8}\) is

\(\frac{8^4 + 8^{16}}{4^8 + 16^8} = \frac{2^{12} + 2^{48} }{ 2^{16} + 2^{32}} = \frac{2^{12} (1 + 2^{36})}{ 2^{16} (1 + 2^{16})} = \frac{1 + 2^{36}}{ 2^4 (1 + 2^{16})} = \frac{2^{36}}{2^4*2^{16}} = 2^{16} = 64*1024 > 50,000 \)

IMO E
­I udnerstand up until this part - what happened to the "1" in the first row?
Show more
­
That approach approximates at that stage:

In the numerator \(1 + 2^{36} \approx 2^{36}\) and in the denominator \(1 + 2^{16}\approx 2^{16}\).

Check here: 

the-value-of-8-4-8-16-4-8-16-8-is-423684.html#p3333997­
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The value of (8^4 + 8^16)/(4^8 + 16^8) is 17 Jul 2025, 05:50
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