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 $3^{2}$.
$$\frac{3\sqrt{3}\sqrt{50}}{\sqrt{54}}$$
Factor $50=5^{2}\times 2$. Rewrite the square root of the product $\sqrt{5^{2}\times 2}$ as the product of square roots $\sqrt{5^{2}}\sqrt{2}$. Take the square root of $5^{2}$.
$$\frac{3\sqrt{3}\times 5\sqrt{2}}{\sqrt{54}}$$
Multiply $3$ and $5$ to get $15$.
$$\frac{15\sqrt{3}\sqrt{2}}{\sqrt{54}}$$
To multiply $\sqrt{3}$ and $\sqrt{2}$, multiply the numbers under the square root.
$$\frac{15\sqrt{6}}{\sqrt{54}}$$
Factor $54=3^{2}\times 6$. Rewrite the square root of the product $\sqrt{3^{2}\times 6}$ as the product of square roots $\sqrt{3^{2}}\sqrt{6}$. Take the square root of $3^{2}$.
$$\frac{15\sqrt{6}}{3\sqrt{6}}$$
Cancel out $3\sqrt{6}$ in both numerator and denominator.
