Completely simplify and rationalize the denominator:
\[\frac{\sqrt{160}}{\sqrt{252}}\times\frac{\sqrt{245}}{\sqrt{128}}\]
+515 \(\frac{\sqrt{160}}{\sqrt{252}}\cdot \frac{\sqrt{245}}{\sqrt{128}}\)
\(=\frac{7\sqrt{5}}{8\sqrt{2}}\cdot \frac{2\sqrt{10}}{3\sqrt{7}}\)
\(=\frac{2\sqrt{10}\cdot 7\sqrt{5}}{3\sqrt{7}\cdot 8\sqrt{2}}\)
\(=\frac{14\sqrt{10}\sqrt{5}}{24\sqrt{7}\sqrt{2}}\)
\(=\frac{\sqrt{7}\sqrt{10}\sqrt{5}}{12\sqrt{2}}\)
\(=\frac{\sqrt{7}\sqrt{2}\sqrt{5}\sqrt{5}}{2^2\cdot \:3\sqrt{2}}\)
\(=\frac{5\sqrt{7}}{3\cdot \:2\sqrt{2}\sqrt{2}}\)
\(=\frac{5\sqrt{7}}{12}\)
pray that i didn't mess up
+515 \(\frac{\sqrt{160}}{\sqrt{252}}\cdot \frac{\sqrt{245}}{\sqrt{128}}\)
\(=\frac{7\sqrt{5}}{8\sqrt{2}}\cdot \frac{2\sqrt{10}}{3\sqrt{7}}\)
\(=\frac{2\sqrt{10}\cdot 7\sqrt{5}}{3\sqrt{7}\cdot 8\sqrt{2}}\)
\(=\frac{14\sqrt{10}\sqrt{5}}{24\sqrt{7}\sqrt{2}}\)
\(=\frac{\sqrt{7}\sqrt{10}\sqrt{5}}{12\sqrt{2}}\)
\(=\frac{\sqrt{7}\sqrt{2}\sqrt{5}\sqrt{5}}{2^2\cdot \:3\sqrt{2}}\)
\(=\frac{5\sqrt{7}}{3\cdot \:2\sqrt{2}\sqrt{2}}\)
\(=\frac{5\sqrt{7}}{12}\)
pray that i didn't mess up
