Question

Find the square root of following surds:
$5+\sqrt{21}$

Answer 1

Let $\sqrt{5+\sqrt{21}}=\pm (\sqrt{a}+\sqrt{b})$

Squaring both sides,

$5+\sqrt{21}=a+b+2\sqrt{ab}$

$a+b=5$ ............. $\left(1\right)$ and $2\sqrt{ab}=\sqrt{21}$

or $4ab=21\Rightarrow a=\frac{21}{4b}$

or, $\frac{21}{4b}+b=5$

or, $4b^2-20b+21=0$

or, $4b^2-6b-14b+21=0$

or, $2b(2b-3)-7(2b-3)=0$

or, $(2b-3)(2b-7)=0$

$\therefore b=\frac{3}{2}$ $b=\frac{7}{2}$

$a=\frac{7}{2}$ $a=\frac{3}{2}$

$\therefore \sqrt{5+\sqrt{21}}=\pm (\sqrt{\frac{3}{2}}+\sqrt{\frac{7}{2}})$ Ans