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 $2^{2}$.
$$\frac{15\times 2\sqrt{2}+9}{15\sqrt{8}+8}$$
Multiply $15$ and $2$ to get $30$.
$$\frac{30\sqrt{2}+9}{15\sqrt{8}+8}$$
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 $2^{2}$.
$$\frac{30\sqrt{2}+9}{15\times 2\sqrt{2}+8}$$
Multiply $15$ and $2$ to get $30$.
$$\frac{30\sqrt{2}+9}{30\sqrt{2}+8}$$
Rationalize the denominator of $\frac{30\sqrt{2}+9}{30\sqrt{2}+8}$ by multiplying numerator and denominator by $30\sqrt{2}-8$.
Consider $\left(30\sqrt{2}+8\right)\left(30\sqrt{2}-8\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.
