Question

rationalize root 2 minus 3 root 5 upon 2 root 2 + 1 root 5

Answer 1

$\frac{ \sqrt{2} - 3 \sqrt{5} }{2 \sqrt{2} + 1 \sqrt{5} } \\ \\ = \frac{ \sqrt{2} - 3 \sqrt{5} }{2 \sqrt{2} + 1 \sqrt{5} } \times \frac{2 \sqrt{2} - 1 \sqrt{5} }{2 \sqrt{2} - 1 \sqrt{5} } \\ \\ = \frac{ \sqrt{2}(2 \sqrt{2} - 1 \sqrt{5} ) - 3 \sqrt{5}(2 \sqrt{2 } - 1 \sqrt{5)} }{ {(2 \sqrt{2)} }^{2} - {(1 \sqrt{5)} }^{2} } \\ \\ = \frac{4 - 1 \sqrt{10} - 6 \sqrt{10} + 15}{8 - 5} \\ \\ = \frac{19 - 7 \sqrt{10} }{3}$

Hope it helps.