Factor $24=2\times 12$. Rewrite the square root of the product $\sqrt{2\times 12}$ as the product of square roots $\sqrt{2}\sqrt{12}$.
$$3\sqrt{2}\sqrt{12}\sqrt{27}\sqrt{2}$$
Multiply $\sqrt{2}$ and $\sqrt{2}$ to get $2$.
$$3\times 2\sqrt{27}\sqrt{12}$$
Multiply $3$ and $2$ to get $6$.
$$6\sqrt{27}\sqrt{12}$$
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 $3^{2}$.
$$6\times 3\sqrt{3}\sqrt{12}$$
Multiply $6$ and $3$ to get $18$.
$$18\sqrt{3}\sqrt{12}$$
Factor $12=2^{2}\times 3$. Rewrite the square root of the product $\sqrt{2^{2}\times 3}$ as the product of square roots $\sqrt{2^{2}}\sqrt{3}$. Take the square root of $2^{2}$.
