To divide \( 9\sqrt{33} \) by \( 27\sqrt{11} \), we can write the expression as follows:
\[ \frac{9\sqrt{33}}{27\sqrt{11}} \]
We can simplify this fraction step by step.
- Simplify the coefficients: The coefficients \( 9 \) and \( 27 \) can be simplified:
\[ \frac{9}{27} = \frac{1}{3} \]
- Combine the square roots: Now we can rewrite the fraction with the square roots:
\[ \frac{\sqrt{33}}{\sqrt{11}} \]
- Simplify the square roots: We can simplify this further:
\[ \frac{\sqrt{33}}{\sqrt{11}} = \sqrt{\frac{33}{11}} = \sqrt{3} \]
Putting it all together, we have:
\[ \frac{9\sqrt{33}}{27\sqrt{11}} = \frac{1}{3} \cdot \sqrt{3} \]
Thus, the final simplified result is:
\[ \frac{\sqrt{3}}{3} \]
So the exact answer is:
\[ \frac{\sqrt{3}}{3} \]
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