Factor $8=2^{2}\times 2$. Rewrite the square root of the product $\sqrt{2^{2}\times 2}$ as the product of square roots $\sqrt{2^{2}}\sqrt{2}$. Take the square root of $2^{2}$.
$$\frac{\sqrt{7}+2\sqrt{2}}{9+2\sqrt{14}}$$
Rationalize the denominator of $\frac{\sqrt{7}+2\sqrt{2}}{9+2\sqrt{14}}$ by multiplying numerator and denominator by $9-2\sqrt{14}$.
Consider $\left(9+2\sqrt{14}\right)\left(9-2\sqrt{14}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.
