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16 Sep 2014, 01:50
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E
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Difficulty:
45%
(medium)
Question Stats:
60% (01:43) correct
40%
(02:35)
wrong
based on 53
sessions
History
Date
Time
Result
Not Attempted Yet
\((\sqrt{7 + \sqrt{48}} + \sqrt{7 - \sqrt{48}})^2 =\)
A. \(1\)
B. \(7 - 4\sqrt{3}\)
C. \(14 - 4\sqrt{3}\)
D. \(14\)
E. \(16\)
A. \(1\)
B. \(7 - 4\sqrt{3}\)
C. \(14 - 4\sqrt{3}\)
D. \(14\)
E. \(16\)
16 Sep 2014, 01:50
Kudos
Bookmarks
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
\((\sqrt{7 + \sqrt{48}} + \sqrt{7 - \sqrt{48}})^2 =\)
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
