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S95-26

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S95-26  [#permalink]

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New post 16 Sep 2014, 01:50
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

59% (01:17) correct 41% (01:43) wrong based on 39 sessions

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Re S95-26  [#permalink]

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New post 16 Sep 2014, 01:50
Official Solution:

\((\sqrt{7 + \sqrt{48}} + \sqrt{7 - \sqrt{48}})^2 =\)

A. \(1\)
B. \(7 - 4\sqrt{3}\)
C. \(14 - 4\sqrt{3}\)
D. \(14\)
E. \(16\)


The question asks for the value of an expression. This expression is in the form of \((x + y)^2\), where

\(x = \sqrt{7 + \sqrt{48}}\) and \(y = \sqrt{7 - \sqrt{48}}\).

The expanded form of \((x + y)^2\) is \(x^2 + 2xy + y^2\). If we substitute our values for \(x\) and \(y\), we get:

\((\sqrt{7 + \sqrt{48}})^2 + 2(\sqrt{7 + \sqrt{48}})(\sqrt{7 - \sqrt{48}}) + (\sqrt{7 - \sqrt{48}})^2\)

We simplify, recalling that the product of two radical terms can be rewritten under one radical sign:

\((7 + \sqrt{48}) + 2\leftarrow(\sqrt{(7 + \sqrt{48})(7 - \sqrt{48})}\rightarrow) + (7 - \sqrt{48})\).

The middle term is the factored form of the difference of two squares, \((x + y)(x - y) = x^2 - y^2\). Simplify accordingly:

\((7 + \sqrt{48}) + 2\leftarrow(\sqrt{(49 - 48)}\rightarrow) + (7 - \sqrt{48})\)

Simplify further, noting that the \(\sqrt{48}\) terms cancel:

\(7 + 7 + 2\sqrt{1} = 14 + 2 = 16\)


Answer: E
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Re: S95-26  [#permalink]

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New post 31 Aug 2017, 07:18
Very good question
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Re: S95-26   [#permalink] 31 Aug 2017, 07:18
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