Factor $75=5^{2}\times 3$. Rewrite the square root of the product $\sqrt{5^{2}\times 3}$ as the product of square roots $\sqrt{5^{2}}\sqrt{3}$. Take the square root of $5^{2}$.
$$\frac{15}{5\sqrt{3}}+\sqrt{108}+\sqrt{432}$$
Rationalize the denominator of $\frac{15}{5\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.
Cancel out $3\times 5$ in both numerator and denominator.
$$\sqrt{3}+\sqrt{108}+\sqrt{432}$$
Factor $108=6^{2}\times 3$. Rewrite the square root of the product $\sqrt{6^{2}\times 3}$ as the product of square roots $\sqrt{6^{2}}\sqrt{3}$. Take the square root of $6^{2}$.
$$\sqrt{3}+6\sqrt{3}+\sqrt{432}$$
Combine $\sqrt{3}$ and $6\sqrt{3}$ to get $7\sqrt{3}$.
$$7\sqrt{3}+\sqrt{432}$$
Factor $432=12^{2}\times 3$. Rewrite the square root of the product $\sqrt{12^{2}\times 3}$ as the product of square roots $\sqrt{12^{2}}\sqrt{3}$. Take the square root of $12^{2}$.
$$7\sqrt{3}+12\sqrt{3}$$
Combine $7\sqrt{3}$ and $12\sqrt{3}$ to get $19\sqrt{3}$.
