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 $3^{2}$.
$$48-\frac{3\sqrt{3}}{2}+\frac{1}{\sqrt{12}}$$
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 $2^{2}$.
$$48-\frac{3\sqrt{3}}{2}+\frac{1}{2\sqrt{3}}$$
Rationalize the denominator of $\frac{1}{2\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.
