Reduce the fraction $\frac{33}{6}$ to lowest terms by extracting and canceling out $3$.
$$\frac{6}{\sqrt{\frac{11}{2}+\frac{5}{9}}}$$
Least common multiple of $2$ and $9$ is $18$. Convert $\frac{11}{2}$ and $\frac{5}{9}$ to fractions with denominator $18$.
$$\frac{6}{\sqrt{\frac{99}{18}+\frac{10}{18}}}$$
Since $\frac{99}{18}$ and $\frac{10}{18}$ have the same denominator, add them by adding their numerators.
$$\frac{6}{\sqrt{\frac{99+10}{18}}}$$
Add $99$ and $10$ to get $109$.
$$\frac{6}{\sqrt{\frac{109}{18}}}$$
Rewrite the square root of the division $\sqrt{\frac{109}{18}}$ as the division of square roots $\frac{\sqrt{109}}{\sqrt{18}}$.
$$\frac{6}{\frac{\sqrt{109}}{\sqrt{18}}}$$
Factor $18=3^{2}\times 2$. Rewrite the square root of the product $\sqrt{3^{2}\times 2}$ as the product of square roots $\sqrt{3^{2}}\sqrt{2}$. Take the square root of $3^{2}$.
$$\frac{6}{\frac{\sqrt{109}}{3\sqrt{2}}}$$
Rationalize the denominator of $\frac{\sqrt{109}}{3\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.
