The value of \(\sqrt {\frac{{\left( {\sqrt {12} - \sqrt 8 } \right)\left( {\sqrt 3 + \sqrt 2 } \right)}}{{5 + \sqrt {24} }}}\)
Solution
Correct Answer: Option E
Numerator :
\(= \sqrt {{\rm{}}\left( {\sqrt {12} {\rm{}} - {\rm{}}\sqrt 8 } \right)\left( {{\rm{}}\sqrt 3 {\rm{}} + {\rm{}}\sqrt 2 } \right)}\)
\(\begin{array}{l} = \sqrt {2{\rm{}} \times \left( {\sqrt 3 - \sqrt 2 } \right) \times \left( {\sqrt 3 {\rm{}} + {\rm{}}\sqrt 2 } \right){\rm{}}} \\ = \sqrt {2 \times \left[ {{\rm{}}{{\left( {\sqrt 3 } \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}{\rm{}}} \right]{\rm{}}} \\ = \sqrt {2{\rm{}} \times {\rm{}}\left( {3{\rm{}} - {\rm{}}2} \right){\rm{}}} \\ = \sqrt {2 \times 1} \\ = \sqrt 2 \end{array}\)
So, we have \(\frac{{\sqrt 2 }}{{\sqrt {5{\rm{}} + {\rm{}}\sqrt {24} } }} = \sqrt 2 \frac{{\sqrt {5{\rm{}} - {\rm{}}\sqrt {24} } {\rm{}}}}{{25 - 24}}\) (Rationalizing the denominator)
\(\sqrt {10{\rm{}} - {\rm{}}4\sqrt 6 }\)
\(\sqrt {4 + 6{\rm{}} - {\rm{}}2 \times 2 \times \sqrt 6 }\)
\(\sqrt {\left( \sqrt 6 - 2\right)^2 }\)
\((\sqrt 6 - 2)\) (Negative value is not considered)
Which is not among the given options.
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