Rewrite the square root of the division $\sqrt{\frac{19}{12}}$ as the division of square roots $\frac{\sqrt{19}}{\sqrt{12}}$.
$$\frac{\sqrt{19}}{\sqrt{12}}+\frac{3}{2}$$
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 $2^{2}$.
$$\frac{\sqrt{19}}{2\sqrt{3}}+\frac{3}{2}$$
Rationalize the denominator of $\frac{\sqrt{19}}{2\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.
To multiply $\sqrt{19}$ and $\sqrt{3}$, multiply the numbers under the square root.
$$\frac{\sqrt{57}}{2\times 3}+\frac{3}{2}$$
Multiply $2$ and $3$ to get $6$.
$$\frac{\sqrt{57}}{6}+\frac{3}{2}$$
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6$ and $2$ is $6$. Multiply $\frac{3}{2}$ times $\frac{3}{3}$.
$$\frac{\sqrt{57}}{6}+\frac{3\times 3}{6}$$
Since $\frac{\sqrt{57}}{6}$ and $\frac{3\times 3}{6}$ have the same denominator, add them by adding their numerators.
