To simplify and rationalize the expression \(\frac{7}{\sqrt{7} - \sqrt{11}}\), we can multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{7} + \sqrt{11}\).
Let's proceed with the calculation:
\[ \frac{7}{\sqrt{7} - \sqrt{11}} \cdot \frac{\sqrt{7} + \sqrt{11}}{\sqrt{7} + \sqrt{11}} = \frac{7(\sqrt{7} + \sqrt{11})}{(\sqrt{7} - \sqrt{11})(\sqrt{7} + \sqrt{11})} \]
Now, calculate the denominator:
\[ (\sqrt{7} - \sqrt{11})(\sqrt{7} + \sqrt{11}) = \sqrt{7}^2 - \sqrt{11}^2 = 7 - 11 = -4 \]
Now, plug this back into the expression:
\[ \frac{7(\sqrt{7} + \sqrt{11})}{-4} \]
This simplifies to:
\[ -\frac{7(\sqrt{7} + \sqrt{11})}{4} \]
So the final simplified form of the expression \(\frac{7}{\sqrt{7} - \sqrt{11}}\), with a rationalized denominator, is:
\[ -\frac{7(\sqrt{7} + \sqrt{11})}{4} \]