Rewrite the square root of the division $\sqrt{\frac{27}{8}}$ as the division of square roots $\frac{\sqrt{27}}{\sqrt{8}}$.
$$\frac{\sqrt{27}}{\sqrt{8}}$$
Factor $27=3^{2}\times 3$. Rewrite the square root of the product $\sqrt{3^{2}\times 3}$ as the product of square roots $\sqrt{3^{2}}\sqrt{3}$. Take the square root of $3^{2}$.
$$\frac{3\sqrt{3}}{\sqrt{8}}$$
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 $2^{2}$.
$$\frac{3\sqrt{3}}{2\sqrt{2}}$$
Rationalize the denominator of $\frac{3\sqrt{3}}{2\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.
