Question

Find the square root of the surd $9+2\sqrt{14}$

Answer 1

Find the square root of the given surd:

Given surd, $9+2\sqrt{14}$

Taking square root we get

$$\begin{gathered} =\sqrt{9+2\sqrt{14}} \\ =\sqrt{9+2\sqrt{7}\sqrt{2}} \\ =\sqrt{7+2\sqrt{7}\sqrt{2}+2} \\ =\sqrt{{\left(\sqrt{7}\right)}^2+2\sqrt{7}\sqrt{2}+{\left(\sqrt{2}\right)}^2} \\ =\sqrt{{\left(\sqrt{7}+\sqrt{2}\right)}^2}\left[\because a^2+2ab+b^2={\left(a+b\right)}^2\right] \\ =\pm \left(\sqrt{7}+\sqrt{2}\right)\mathbf{}\left[\because \sqrt{a^2}=\pm a\right] \end{gathered}$$

Hence, the square root of $9+2\sqrt{14}$ is $\pm \left(\sqrt{7}+\sqrt{2}\right)$.