Rationalize \(\frac{5\sqrt{7} - 7\sqrt{5}}{\sqrt{7} - \sqrt{5}}\)
A. -2 \sqrt{35}\)
B. 4\sqrt{7}\) - 6\(\sqrt{5}\)
C. -\(\sqrt{35}\)
D. 4\sqrt{7}\) - 8\(\sqrt{5}\)
E. \(\sqrt{35}\)
Correct Answer: C
Explanation
\(\frac{5\sqrt{7} - 7\sqrt{5}}{\sqrt{7} - \sqrt{5}}\) = \(\frac{5\sqrt{7} - 7\sqrt{5}}{\sqrt{7} - \sqrt{5}}\) x \(\frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}\)
= \(\frac{(5 \times 7) + (5 \sqrt{7} \times 5) - (7 \times \sqrt{5} \times 7) (-7 \times 5)}{(\sqrt{7})^2}\)
= \(\frac{5 \sqrt{35} - 7\sqrt{35}}{2}\)
= \(\frac{-2\sqrt{35}}{2}\)
= - \(\sqrt{35}\)
