To rationalize the denominator of the expression \(\frac{13}{3 + \sqrt{11}}\), you can multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 - \sqrt{11}\):
\[ \frac{13}{3 + \sqrt{11}} \cdot \frac{3 - \sqrt{11}}{3 - \sqrt{11}} \]
Now, multiply the numerators and the denominators:
Numerator: \[ 13(3 - \sqrt{11}) = 39 - 13\sqrt{11} \]
Denominator: Using the difference of squares: \[ (3 + \sqrt{11})(3 - \sqrt{11}) = 3^2 - (\sqrt{11})^2 = 9 - 11 = -2 \]
Now, combining both results gives:
\[ \frac{39 - 13\sqrt{11}}{-2} \]
This can be simplified by splitting up the terms in the numerator:
\[ \frac{39}{-2} - \frac{13\sqrt{11}}{-2} = -\frac{39}{2} + \frac{13\sqrt{11}}{2} \]
Thus, we have:
\[ -\frac{39}{2} + \frac{13\sqrt{11}}{2} = \frac{13\sqrt{11} - 39}{2} \]
So the final answer, with the denominator rationalized, is:
\[ \boxed{\frac{13\sqrt{11} - 39}{2}} \]
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