To divide \( 9 \sqrt{33} \) by \( 27 \sqrt{11} \), we can set up the expression as follows:
\[ \frac{9 \sqrt{33}}{27 \sqrt{11}} \]
Next, we can simplify this fraction. First, we can divide the coefficients (the numbers in front of the square roots):
\[ \frac{9}{27} = \frac{1}{3} \]
Now, we can rewrite the expression as:
\[ \frac{1}{3} \cdot \frac{\sqrt{33}}{\sqrt{11}} \]
Next, we can simplify the square root fraction:
\[ \frac{\sqrt{33}}{\sqrt{11}} = \sqrt{\frac{33}{11}} = \sqrt{3} \]
Combining these results gives us:
\[ \frac{1}{3} \sqrt{3} \]
Therefore, the exact answer is:
\[ \frac{\sqrt{3}}{3} \]