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M25-36

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Bunuel
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M25-36 [#permalink] New post   16 Sep 2014, 01:24
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Which of the following is equal to \(\frac{1}{2 - \sqrt{3}}\)?

A. \(\sqrt{3} - 2\)
B. \(2 + \sqrt{3}\)
C. \(\sqrt{2} + \sqrt{3}\)
D. \(2 - \sqrt{3}\)
E. \(\sqrt{3} + 4\)
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B
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Bunuel
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Re M25-36 [#permalink] New post   16 Sep 2014, 01:24
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Official Solution:

Which of the following is equal to \(\frac{1}{2 - \sqrt{3}}\)?

A. \(\sqrt{3} - 2\)
B. \(2 + \sqrt{3}\)
C. \(\sqrt{2} + \sqrt{3}\)
D. \(2 - \sqrt{3}\)
E. \(\sqrt{3} + 4\)


Rationalize the fraction by multiplying both the numerator and the denominator by \((2 + \sqrt{3})\). The purpose of rationalization is to eliminate irrational expressions in the denominator and simplify the expression:

\(\frac{2 + \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \)

\(=\frac{2 + \sqrt{3}}{4 - 3} = \)

\(=2 + \sqrt{3}\).


Answer: B
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Re: M25-36 [#permalink] New post   12 Oct 2014, 09:52
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Bunuel wrote:
Official Solution:

\(\frac{1}{2 - \sqrt{3}} = ?\)

A. \(\sqrt{3} - 2\)
B. \(2 + \sqrt{3}\)
C. \(\sqrt{2} + \sqrt{3}\)
D. \(2 - \sqrt{3}\)
E. \(\sqrt{3} + 4\)


Multiply both numerator and denominator by \((2 + \sqrt{3})\). That yields \(\frac{2 + \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3}\).


Answer: B


Bunuel,

I'm just curious about what the intuition is behind this type of solution. How do we know we should multiply by \((2 + \sqrt{3})\) as opposed to some other expression, or rather than just solving every answer in raw form?
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Bunuel
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Re: M25-36 [#permalink] New post   12 Oct 2014, 12:02
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bsmith37 wrote:
Bunuel wrote:
Official Solution:

\(\frac{1}{2 - \sqrt{3}} = ?\)

A. \(\sqrt{3} - 2\)
B. \(2 + \sqrt{3}\)
C. \(\sqrt{2} + \sqrt{3}\)
D. \(2 - \sqrt{3}\)
E. \(\sqrt{3} + 4\)


Multiply both numerator and denominator by \((2 + \sqrt{3})\). That yields \(\frac{2 + \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3}\).


Answer: B


Bunuel,

I'm just curious about what the intuition is behind this type of solution. How do we know we should multiply by \((2 + \sqrt{3})\) as opposed to some other expression, or rather than just solving every answer in raw form?


This algebraic manipulation is called rationalisation and is performed to eliminate irrational expression in the denominator. For this particular case we are doing rationalisation by applying the following rule: \((a-b)(a+b)=a^2-b^2\). The concept of rationalisation of a fraction used to solve this problem is often tested on the GMAT.

Questions relied on this technique:
if-x-0-then-106291.html
if-n-is-positive-which-of-the-following-is-equal-to-31236.html
consider-a-quarter-of-a-circle-of-radius-16-let-r-be-the-131083.html
in-the-diagram-not-drawn-to-scale-sector-pq-is-a-quarter-139282.html
in-the-diagram-what-is-the-value-of-x-129962.html
the-perimeter-of-a-right-isoscles-triangle-is-127049.html
which-of-the-following-is-equal-to-98531.html
if-x-is-positive-then-1-root-x-1-root-x-163491.html

Hope it helps.
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Re: M25-36 [#permalink] New post   17 Oct 2023, 00:05
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M25-36 [#permalink]
17 Oct 2023, 00:05
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