Question

Square root of $\sqrt{5-\sqrt{21}}$.

Answer 1

Let $\sqrt{5-\sqrt{21}}=\pm (\sqrt{x}-\sqrt{y})......\left(1\right)$

Squaring both sides,

$\Rightarrow 5-\sqrt{21}=x+y-2\sqrt{xy}$

Comparing both sides,

$\Rightarrow x+y=\mathrm{5........}\left(2\right)$

and $2\sqrt{xy}=\sqrt{21}$

Squaring both sides,

$\Rightarrow 4xy=21$

$\Rightarrow xy=\frac{21}{4}......\left(3\right)$

Now, squaring both sides of eq$\left(1\right)$

${(x+y)}^2=25$

$\Rightarrow x^2+y^2+2xy=25$

$\Rightarrow x^2+y^2+2\times \frac{21}{4}=25$

$\Rightarrow x^2+y^2=25-\frac{21}{2}$

$\Rightarrow x^2+y^2=\frac{29}{2}.......\left(4\right)$

Now,${(x-y)}^2=x^2+y^2-2xy=\frac{29}{2}-\frac{21}{2}=\frac{8}{2}=4$

$\Rightarrow x-y=\mathrm{2......}\left(5\right)$

Solving $\left(2\right)$ and $\left(5\right)$

$x=\frac{7}{2},y=\frac{3}{2}$

From $\left(1\right)$

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

Hence, solved.