Question

(\frac{4 + sqrt{2}}{sqrt{5} + sqrt{7}})

Explanation:

To rationalize the denominator of the fraction \(\frac{4 + \sqrt{2}}{\sqrt{5}+\sqrt{7}}\), we can use the conjugate of the denominator. The conjugate of \(\sqrt{5}+\sqrt{7}\) is \(\sqrt{5}-\sqrt{7}\). We multiply the numerator and the denominator by the conjugate:

\[

$$\begin{align*} \frac{4 + \sqrt{2}}{\sqrt{5}+\sqrt{7}}&=\frac{(4 + \sqrt{2})(\sqrt{5}-\sqrt{7})}{(\sqrt{5}+\sqrt{7})(\sqrt{5}-\sqrt{7})}\\ &=\frac{4\sqrt{5}-4\sqrt{7}+\sqrt{2}\sqrt{5}-\sqrt{2}\sqrt{7}}{(\sqrt{5})^2-(\sqrt{7})^2}\\ &=\frac{4\sqrt{5}-4\sqrt{7}+\sqrt{10}-\sqrt{14}}{5 - 7}\\ &=\frac{4\sqrt{5}-4\sqrt{7}+\sqrt{10}-\sqrt{14}}{-2}\\ &=\frac{-4\sqrt{5}+4\sqrt{7}-\sqrt{10}+\sqrt{14}}{2}\\ &=- 2\sqrt{5}+2\sqrt{7}-\frac{\sqrt{10}}{2}+\frac{\sqrt{14}}{2} \end{align*}$$

\]

Answer:

\(- 2\sqrt{5}+2\sqrt{7}-\frac{\sqrt{10}}{2}+\frac{\sqrt{14}}{2}\) (or the equivalent form \(\frac{-4\sqrt{5} + 4\sqrt{7}-\sqrt{10}+\sqrt{14}}{2}\))