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

Question

Accepted Answer

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 $\frac{-4\sqrt{5} + 4\sqrt{7}-\sqrt{10}+\sqrt{14}}{2}$)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: