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Basic of Logarithms
medium
$\sqrt {(3 + \sqrt 5 )} = . .$ .
A
$\sqrt 5 + 1$
B
$\sqrt 3 + \sqrt 2 $
C
$(\sqrt 5 + 1)/\sqrt 2 $
D
${1 \over 2}(\sqrt 5 + 1)$
Solution
(c) Let $\sqrt {3 + \sqrt 5 } = \sqrt x + \sqrt y $
$3 + \sqrt 5 = \,x + y + 2\sqrt {xy} $. Obviously $x + y = 3$
and $4xy = 5$. So ${(x – y)^2} = 9 – 5 = 4$ or $(x – y) = 2$
After solving $x = {5 \over 2},y = {1 \over 2}$.
Hence, $\sqrt {3 + \sqrt 5 } = \sqrt {{5 \over 2}} + \sqrt {{1 \over 2}} = {{\sqrt 5 + 1} \over {\sqrt 2 }}$.
Standard 11
Mathematics
