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 $3^{2}$.
$$\frac{5\times 3\sqrt{3}}{3}\sqrt{24}$$
Multiply $5$ and $3$ to get $15$.
$$\frac{15\sqrt{3}}{3}\sqrt{24}$$
Divide $15\sqrt{3}$ by $3$ to get $5\sqrt{3}$.
$$5\sqrt{3}\sqrt{24}$$
Factor $24=2^{2}\times 6$. Rewrite the square root of the product $\sqrt{2^{2}\times 6}$ as the product of square roots $\sqrt{2^{2}}\sqrt{6}$. Take the square root of $2^{2}$.
$$5\sqrt{3}\times 2\sqrt{6}$$
Multiply $5$ and $2$ to get $10$.
$$10\sqrt{3}\sqrt{6}$$
Factor $6=3\times 2$. Rewrite the square root of the product $\sqrt{3\times 2}$ as the product of square roots $\sqrt{3}\sqrt{2}$.
