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:

$$
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$$

Answer:

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