To rationalize the denominator of the expression \( \frac{7}{2 + \sqrt{13}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( 2 - \sqrt{13} \).
So, we have:
\[ \frac{7}{2 + \sqrt{13}} \cdot \frac{2 - \sqrt{13}}{2 - \sqrt{13}} = \frac{7(2 - \sqrt{13})}{(2 + \sqrt{13})(2 - \sqrt{13})} \]
Let's calculate the denominator first:
\[ (2 + \sqrt{13})(2 - \sqrt{13}) = 2^2 - (\sqrt{13})^2 = 4 - 13 = -9 \]
Now, let's calculate the numerator:
\[ 7(2 - \sqrt{13}) = 14 - 7\sqrt{13} \]
Putting it all together, we get:
\[ \frac{14 - 7\sqrt{13}}{-9} = -\frac{14 - 7\sqrt{13}}{9} = -\frac{14}{9} + \frac{7\sqrt{13}}{9} \]
Thus, the rationalized form of the expression is:
\[ -\frac{14}{9} + \frac{7\sqrt{13}}{9} \]
This can also be written as:
\[ \frac{7\sqrt{13}}{9} - \frac{14}{9} \]
So, your final answer is:
\[ \frac{7\sqrt{13}}{9} - \frac{14}{9} \]