Question
The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$ is equal to
A.
$$1 - \sqrt 5 + \sqrt 2 + \sqrt {10} $$
B.
$$1 + \sqrt 5 + \sqrt 2 - \sqrt {10} $$
C.
$$1 + \sqrt 5 - \sqrt 2 + \sqrt {10} $$
D.
$$1 - \sqrt 5 - \sqrt 2 + \sqrt {10} $$
Answer :
$$1 + \sqrt 5 + \sqrt 2 - \sqrt {10} $$
Solution :
$$\eqalign{
& \frac{{12}}{{\left( {3 + \sqrt 5 } \right) + 2\sqrt 2 }} \cr
& = \frac{{12}}{{\left( {3 + \sqrt 5 } \right) + 2\sqrt 2 }} \times \frac{{\left( {3 + \sqrt 5 } \right) - 2\sqrt 2 }}{{\left( {3 + \sqrt 5 } \right) - 2\sqrt 2 }} \cr
& = \frac{{12\left[ {3 + \sqrt 5 - 2\sqrt 2 } \right]}}{{{{\left( {3 + \sqrt 5 } \right)}^2} - {{\left( {2\sqrt 2 } \right)}^2}}} \cr
& = \frac{{12\left[ {3 + \sqrt 5 - 2\sqrt 2 } \right]}}{{9 + 5 + 6\sqrt 5 - 8}} \cr
& = \frac{{12\left[ {3 + \sqrt 5 - 2\sqrt 2 } \right]}}{{6\left( {\sqrt 5 + 1} \right)}} \times \frac{{\sqrt 5 - 1}}{{\sqrt 5 - 1}} \cr
& = \frac{{2\left[ {3\sqrt 5 + 5 - 2\sqrt {10} - 3 - \sqrt 5 + 2\sqrt 2 } \right]}}{{5 - 1}} \cr
& = \frac{{2 + 2\sqrt 5 + 2\sqrt 2 - 2\sqrt {10} }}{2} \cr
& = 1 + \sqrt 2 + \sqrt 5 - \sqrt {10} \cr} $$