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Question

Question: \[\frac{12}{3 + \sqrt{5} - 2\sqrt{2}} =\]...

123+5−22=\frac{12}{3 + \sqrt{5} - 2\sqrt{2}} =

A

1+5+(10)+21 + \sqrt{5} + \sqrt{(10)} + \sqrt{2}

B

1+5−(10)+21 + \sqrt{5} - \sqrt{(10)} + \sqrt{2}

C

1+5+10−21 + \sqrt{5} + \sqrt{10} - \sqrt{2}

D

1−5−2+(10)1 - \sqrt{5} - \sqrt{2} + \sqrt{(10)}

Answer

1+5+10−21 + \sqrt{5} + \sqrt{10} - \sqrt{2}

Explanation

Solution

123+5−22=12[(3−22)−5][(3−22)+5][(3−22)−5]\frac{12}{3 + \sqrt{5} - 2\sqrt{2}} = \frac{12\lbrack(3 - 2\sqrt{2}) - \sqrt{5}\rbrack}{\lbrack(3 - 2\sqrt{2}) + \sqrt{5}\rbrack\lbrack(3 - 2\sqrt{2}) - \sqrt{5}\rbrack}

=12(3−22−5)(3−22)2−5=12(3−22−5)17−122−5= \frac{12(3 - 2\sqrt{2} - \sqrt{5})}{(3 - 2\sqrt{2})^{2} - 5} = \frac{12(3 - 2\sqrt{2} - \sqrt{5})}{17 - 12\sqrt{2} - 5}

=(3−22−5)1−2=(5+22−3)(2+1)(2−1)(2+1)= \frac{(3 - 2\sqrt{2} - \sqrt{5})}{1 - \sqrt{2}} = \frac{(\sqrt{5} + 2\sqrt{2} - 3)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)}

=10+4−32+5+22−32−1=1+5+10−2= \frac{\sqrt{10} + 4 - 3\sqrt{2} + \sqrt{5} + 2\sqrt{2} - 3}{2 - 1} = 1 + \sqrt{5} + \sqrt{10} - \sqrt{2}.