\begin{tabular}{l} Question \\ Simplify \( \frac{\sqrt{13}}{\sqrt{z}+\sqrt{15}} \) by rationalizing the denominator. \\ Provide your answer below: \\ \hline\end{tabular}

To simplify \( \frac{\sqrt{13}}{\sqrt{z}+\sqrt{15}} \) by rationalizing the denominator, multiply both the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{z} - \sqrt{15} \). So, we have: \[ \frac{\sqrt{13}(\sqrt{z}-\sqrt{15})}{(\sqrt{z}+\sqrt{15})(\sqrt{z}-\sqrt{15})} \] The denominator simplifies using the difference of squares: \[ (\sqrt{z})^2 - (\sqrt{15})^2 = z - 15 \] Now, simplifying the numerator gives: \[ \sqrt{13}(\sqrt{z} - \sqrt{15}) = \sqrt{13z} - \sqrt{195} \] Putting it all together, the final simplified expression is: \[ \frac{\sqrt{13z} - \sqrt{195}}{z - 15} \]