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Which of the following is the best approximation for 11^{11}-9^9?

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Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 21 Aug 2018, 03:38
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A
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[Math Revolution GMAT math practice question]

Which of the following is the best approximation for \(11^{11}-9^9\)?

A. \(11^8\)
B. \(11^9\)
C. \(11^{10}\)
D. \(11^{11}\)
E. \(11^{12}\)

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Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post Updated on: 21 Aug 2018, 08:20
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

Which of the following is the best approximation for \(11^{11}-9^9\)?

A. \(11^8\)
B. \(11^9\)
C. \(11^{10}\)
D. \(11^{11}\)
E. \(11^{12}\)


Since the answer choices are very spread out, our approximation does not have to very exact.
\(11^{11}-9^9 = 11^{11} - (11-2)^9 ≈ 11^{11} - 11^9 = 11^9(11^2 - 1) ≈ (11^9)(11^2) = 11^{11}\)


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Originally posted by GMATGuruNY on 21 Aug 2018, 04:43.
Last edited by GMATGuruNY on 21 Aug 2018, 08:20, edited 1 time in total.
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Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 21 Aug 2018, 05:01
MathRevolution wrote:
[Math Revolution GMAT math practice question]

Which of the following is the best approximation for \(11^{11}-9^9\)?

A. \(11^8\)
B. \(11^9\)
C. \(11^{10}\)
D. \(11^{11}\)
E. \(11^{12}\)


answer would be \(11^{10}\) if \(9^9>\frac{11^{11}}{2}\), otherwise 11^{11}..
so lets see... \(\frac{11^{11}}{2} = \frac{11}{2}*11^{10}=5.5*11^{10}\) This is much greater than \(9^9\)..
so you can neglect \(9^9\) when compared to \(11^{11}\)
so answer is \(11^{11}\)

D
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Re: Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 21 Aug 2018, 08:15
GMATGuruNY wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

Which of the following is the best approximation for \(11^{11}-9^9\)?

A. \(11^8\)
B. \(11^9\)
C. \(11^{10}\)
D. \(11^{11}\)
E. \(11^{12}\)


Since the answer choices are very spread out, our approximation does not have to very exact.
\(11^{11}-9^9 = 11^{11} - (11-2)^{11} ≈ 11^{11} - 11^9 = 11^9(11^2 - 1) ≈ (11^9)(11^2) = 11^{11}\)



Have been your follower for a long time on beat the gmat!!Big fan of yours..cant afford so your posts are Bible :) .. good to have you here Sir.

I think you mistyped the bold part .(just rewriting)

.
\(11^{11}-9^9 = 11^{11} - (11-2)^{09} ≈ 11^{11} - 11^9 = 11^9(11^2 - 1) ≈ (11^9)(11^2) = 11^{11}\)



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Re: Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 21 Aug 2018, 08:47
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prabsahi wrote:
Have been your follower for a long time on beat the gmat!!Big fan of yours..cant afford so your posts are Bible :) .. good to have you here Sir.

I think you mistyped the bold part .(just rewriting)

.
\(11^{11}-9^9 = 11^{11} - (11-2)^{09} ≈ 11^{11} - 11^9 = 11^9(11^2 - 1) ≈ (11^9)(11^2) = 11^{11}\)



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Fixed!
Thanks for catching the typo.

Another way to approach this problem:
10¹¹ (100,000,000,000) is 100 TIMES the value of 10⁹ (1,000,000,000).
Implication:
11¹¹ must be MORE THAN 100 TIMES the value of 9⁹.
Since 9⁹ is so small when compared to 11¹¹, we can ignore the value of 9⁹ when we approximate:
11¹¹ - 9⁹ ≈ 11¹¹


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Re: Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 23 Aug 2018, 02:12
=>

\(9^9\) is much smaller than \(11^{11}\).
Thus, \(11^{11}-9^9\) is closest to \(11^{11}.\)

Therefore, the answer is D.
Answer: D
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Which of the following is the best approximation for 11^{11}-9^9?  [#permalink]

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New post 12 Jan 2019, 04:38
I think the easiest way goes like this

\(11^(11)-9^9 ≈ 10^(11)-10^9 = 10^9(10^2-1) = 10^9 (10-1) (10+1) ≈ 10^(11)\)
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Which of the following is the best approximation for 11^{11}-9^9?   [#permalink] 12 Jan 2019, 04:38
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Which of the following is the best approximation for 11^{11}-9^9?

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