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Completely simplify and rationalize the denominator:$$\frac{\sqrt{160}}{\sqrt{252}}\times\frac{\sqrt{245}}{\sqrt{75}}$$

 Mar 20, 2022
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\(\sqrt {160} = \sqrt{16} \times \sqrt{10} = {4 \sqrt10}\)

\(\sqrt{252} = \sqrt{36} \times \sqrt7 = 6\sqrt7\)

\(\sqrt{245} =\sqrt{49} \times \sqrt5 = 7\sqrt5\)

\(\sqrt{75} = \sqrt{25} \times \sqrt{3} = 5\sqrt3\)

\({4\sqrt{10} \over 6\sqrt7} \times {7\sqrt5 \over 5\sqrt3} = {28\sqrt{50} \over 30\sqrt{21}}\)

\({28\sqrt{50}\over30\sqrt{21}} = {14\sqrt{50} \over 15\sqrt{21}} = {14 \times \sqrt{25} \times \sqrt2 \over 15\sqrt{21}} = {14 \times 5 \times \sqrt2 \over 15\sqrt{21}} = {70\sqrt2 \over 15\sqrt{21}}={14\sqrt2 \over 3\sqrt{21}}\)

\({14\sqrt2 \over 3\sqrt{21}} \times {3\sqrt{21}\over3\sqrt{21}} = {42\sqrt{42}\over 189} = \color{brown}\boxed{2\sqrt{42}\over 9}\)

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 Mar 20, 2022

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