Simplify; \(\frac{\sqrt{5} + 3}{4 - \sqrt{10}}\)
A. \(\frac{2}{3}\)\(\sqrt{5}\) + \(\frac{5}{6}\sqrt{2}\) + 2
B. \(\frac{2}{3}\)\(\sqrt{5}\) + \(\frac{5}{6}\sqrt{2}\) + \(\frac{1}{2}\sqrt{10}\)
C. \(\frac{2}{3}\)\(\sqrt{5}\) + \(\frac{5}{6}\sqrt{2}\) + \(\frac{1}{2}\sqrt{10}\) + 2
D. \(\frac{2}{3}\)\(\sqrt{5}\) - \(\frac{5}{6}\sqrt{2}\) + \(\frac{1}{2}\sqrt{10}\) + 2
Correct Answer: C
Explanation
\(\frac{(\sqrt{5} + 3)(4 + \sqrt{10})}{(4 - \sqrt{10})(4 + \sqrt{10})}\)
= \(\frac{4\sqrt{5} + \sqrt{50} + 12 + 3\sqrt{10}}{4^2 - (\sqrt{10})^2}\)
= \(\frac{4\sqrt{5} + 5\sqrt{2} + 12 + 3\sqrt{10}}{16 - 10}\)
= \(\frac{4 \sqrt{5}}{6} + \frac{5 \sqrt{2}}{6} + \frac{12}{6} + \frac{3\sqrt{10}}{6}\)
= \(\frac{2}{3}\)\(\sqrt{5}\) + \(\frac{5}{6}\sqrt{2}\) + \(\frac{1}{2}\sqrt{10}\) + 2
