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555-605 (Medium)|   Algebra|      
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Bunuel
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 27 May 2021, 09:00
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25% (medium)

Question Stats:

69% (01:33) correct 31% (01:58) wrong based on 105 sessions

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What is the value of \((\frac{9}{7 + 2\sqrt{10}} + 2\sqrt{10})^3\)?

A. 64
B. 140
C. 216
D. 230
E. 343
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E
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? Updated on: 05 Jun 2021, 09:33
Originally posted by Kinshook on 27 May 2021, 09:30.
Last edited by Kinshook on 05 Jun 2021, 09:33, edited 2 times in total.
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Asked: What is the value of \((\frac{9}{7 + 2\sqrt{10}} + 2\sqrt{10})^3\)?



\((7 + 2\sqrt{10})(7 - 2\sqrt{10}) = 49 - 40 = 9\)
\(\frac{9}{7 + 2\sqrt{10}} = 7 - 2\sqrt{10}\)

\((\frac{9}{7 + 2\sqrt{10}} + 2\sqrt{10})^3 = (7 - 2\sqrt{10} + 2\sqrt{10})^3 = 7^3 = 343\)

IMO E
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 27 May 2021, 11:17
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(9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3
Rationalising:
\(\frac{9(7-2\sqrt{10})}{(7-2\sqrt{10})(7+2\sqrt{10})}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})}{(49-40)}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})+9(2\sqrt{10})}{9}\)
=7^3
=343
IMO E
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 05 Jun 2021, 08:09
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Anisha1637
(9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3
Rationalising:
\(\frac{9(7-2\sqrt{10})}{(7-2\sqrt{10})(7+2\sqrt{10})}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})}{(49-40)}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})+9(2\sqrt{10})}{9}\)
=7^3
=343
IMO E
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Hi! How did you get 49-40?
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 05 Jun 2021, 08:34
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AbhaGanu
Anisha1637
(9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3
Rationalising:
\(\frac{9(7-2\sqrt{10})}{(7-2\sqrt{10})(7+2\sqrt{10})}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})}{(49-40)}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})+9(2\sqrt{10})}{9}\)
=7^3
=343
IMO E

Hi! How did you get 49-40?
Show more

Apply \((a - b)(a + b) = a^2 - b^2\):

\((7-2\sqrt{10})(7+2\sqrt{10})=7^2 - (2\sqrt{10})^2 = 49 -40=9\)
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 06 Jun 2021, 01:21
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Bunuel
AbhaGanu
Anisha1637
(9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3
Rationalising:
\(\frac{9(7-2\sqrt{10})}{(7-2\sqrt{10})(7+2\sqrt{10})}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})}{(49-40)}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})+9(2\sqrt{10})}{9}\)
=7^3
=343
IMO E

Hi! How did you get 49-40?

Apply \((a - b)(a + b) = a^2 - b^2\):

\((7-2\sqrt{10})(7+2\sqrt{10})=7^2 - (2\sqrt{10})^2 = 49 -40=9\)
Show more


Hi Anisha, thank you! But I haven't yet understood how did you arrive at 7^3 from the step before that.
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? Updated on: 08 Jun 2021, 04:01
Originally posted by GmatKnightTutor on 06 Jun 2021, 02:29.
Last edited by GmatKnightTutor on 08 Jun 2021, 04:01, edited 3 times in total.
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 06 Jun 2021, 06:19
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Bunuel
What is the value of \((\frac{9}{7 + 2\sqrt{10}} + 2\sqrt{10})^3\)?

A. 64
B. 140
C. 216
D. 230
E. 343
Show more

When I scan the answer choices (always scan the answer choices before determining the approach you'll take), I see that the answer choices are reasonably spread apart, except for C and D.
Since solving the question algebraically will likely take quite a while, I'm going to try solving the question through approximation and cross my fingers that I don't end up with an answer that's between 216 and 230.

Since \(\sqrt{9} = 3\) and\( \sqrt{16} = 4\), I know that \(\sqrt{10} = 3.something\).
I'm going to guess that it's about \(3.1\)

So let's take the original expression and replace \(\sqrt{9}\) with \(3.1\)

\((\frac{9}{7 + 2\sqrt{10}} + 2\sqrt{10})^3≈(\frac{9}{7 + 2(3.1)} + 2(3.1))^3\)

\(≈(\frac{9}{7 + 6.2} + 6.2)^3\)

\(≈(\frac{9}{13.2} + 6.2)^3\)

\(≈(0.7 + 6.2)^3\)

\(≈(6.9)^3\)

Since \(6^3=216\) and \(7^3=343\), I'll take that's a choice E.
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What is the value of (9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3 ? 07 Jun 2021, 05:13
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Bunuel
AbhaGanu
Anisha1637
(9/(7 + 2*10^(1/2)) + 2*10^(1/2))^3
Rationalising:
\(\frac{9(7-2\sqrt{10})}{(7-2\sqrt{10})(7+2\sqrt{10})}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})}{(49-40)}+2\sqrt{10}\)
=\(\frac{9(7-2\sqrt{10})+9(2\sqrt{10})}{9}\)
=7^3
=343
IMO E

Hi! How did you get 49-40?

Apply \((a - b)(a + b) = a^2 - b^2\):

\((7-2\sqrt{10})(7+2\sqrt{10})=7^2 - (2\sqrt{10})^2 = 49 -40=9\)
Show more

Could you explain why you would use \((a - b)(a + b) = a^2 - b^2\):
I can't figure out how to solve this question
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