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 $2^{2}$.
$$\frac{24}{2\sqrt{6}\sqrt{27}}$$
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 $3^{2}$.
$$\frac{24}{2\sqrt{6}\times 3\sqrt{3}}$$
Multiply $2$ and $3$ to get $6$.
$$\frac{24}{6\sqrt{6}\sqrt{3}}$$
Factor $6=3\times 2$. Rewrite the square root of the product $\sqrt{3\times 2}$ as the product of square roots $\sqrt{3}\sqrt{2}$.
$$\frac{24}{6\sqrt{3}\sqrt{2}\sqrt{3}}$$
Multiply $\sqrt{3}$ and $\sqrt{3}$ to get $3$.
$$\frac{24}{6\times 3\sqrt{2}}$$
Rationalize the denominator of $\frac{24}{6\times 3\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.
