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\frac{\frac{\sqrt{16}}{\sqrt{5}}}{\sqrt{\frac{8}{5}}}
Rewrite the square root of the division \sqrt{\frac{16}{5}} as the division of square roots \frac{\sqrt{16}}{\sqrt{5}}.
\frac{\frac{4}{\sqrt{5}}}{\sqrt{\frac{8}{5}}}
Calculate the square root of 16 and get 4.
\frac{\frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\sqrt{\frac{8}{5}}}
Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.
\frac{\frac{4\sqrt{5}}{5}}{\sqrt{\frac{8}{5}}}
The square of \sqrt{5} is 5.
\frac{\frac{4\sqrt{5}}{5}}{\frac{\sqrt{8}}{\sqrt{5}}}
Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.
\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}}{\sqrt{5}}}
Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.
\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}
Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.
\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}\sqrt{5}}{5}}
The square of \sqrt{5} is 5.
\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{10}}{5}}
To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.
\frac{4\sqrt{5}\times 5}{5\times 2\sqrt{10}}
Divide \frac{4\sqrt{5}}{5} by \frac{2\sqrt{10}}{5} by multiplying \frac{4\sqrt{5}}{5} by the reciprocal of \frac{2\sqrt{10}}{5}.
\frac{2\sqrt{5}}{\sqrt{10}}
Cancel out 2\times 5 in both numerator and denominator.
\frac{2\sqrt{5}\sqrt{10}}{\left(\sqrt{10}\right)^{2}}
Rationalize the denominator of \frac{2\sqrt{5}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.
\frac{2\sqrt{5}\sqrt{10}}{10}
The square of \sqrt{10} is 10.
\frac{2\sqrt{5}\sqrt{5}\sqrt{2}}{10}
Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.
\frac{2\times 5\sqrt{2}}{10}
Multiply \sqrt{5} and \sqrt{5} to get 5.
\frac{10\sqrt{2}}{10}
Multiply 2 and 5 to get 10.
\sqrt{2}
Cancel out 10 and 10.