Solve sqrt{4/3}divsqrt{8/3}*sqrt{8/5}

$$\sqrt { \frac { 4 } { 3 } } \div \sqrt { \frac { 8 } { 3 } } \times \sqrt { \frac { 8 } { 5 } }$$

Evaluate

$$\frac{2\sqrt{5}}{5}\approx 0.894427191$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{4}}{\sqrt{3}}}{\sqrt{\frac{8}{3}}}\sqrt{\frac{8}{5}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{2}{\sqrt{3}}}{\sqrt{\frac{8}{3}}}\sqrt{\frac{8}{5}}$$

Rationalize the denominator of \frac{2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{8}{3}}}\sqrt{\frac{8}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{3}}{3}}{\sqrt{\frac{8}{3}}}\sqrt{\frac{8}{5}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{\sqrt{8}}{\sqrt{3}}}\sqrt{\frac{8}{5}}$$

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}.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{2\sqrt{2}}{\sqrt{3}}}\sqrt{\frac{8}{5}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\sqrt{\frac{8}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{2\sqrt{2}\sqrt{3}}{3}}\sqrt{\frac{8}{5}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{2\sqrt{6}}{3}}\sqrt{\frac{8}{5}}$$

Divide \frac{2\sqrt{3}}{3} by \frac{2\sqrt{6}}{3} by multiplying \frac{2\sqrt{3}}{3} by the reciprocal of \frac{2\sqrt{6}}{3}.

$$\frac{2\sqrt{3}\times 3}{3\times 2\sqrt{6}}\sqrt{\frac{8}{5}}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{\sqrt{3}}{\sqrt{6}}\sqrt{\frac{8}{5}}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{3}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\sqrt{\frac{8}{5}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{3}\sqrt{6}}{6}\sqrt{\frac{8}{5}}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{\sqrt{3}\sqrt{3}\sqrt{2}}{6}\sqrt{\frac{8}{5}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{2}}{6}\sqrt{\frac{8}{5}}$$

Divide 3\sqrt{2} by 6 to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{2}\sqrt{2}\sqrt{\frac{8}{5}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$\frac{1}{2}\sqrt{2}\times \frac{\sqrt{8}}{\sqrt{5}}$$

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}.

$$\frac{1}{2}\sqrt{2}\times \frac{2\sqrt{2}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{1}{2}\sqrt{2}\times \frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{1}{2}\sqrt{2}\times \frac{2\sqrt{2}\sqrt{5}}{5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{1}{2}\sqrt{2}\times \frac{2\sqrt{10}}{5}$$

Multiply \frac{1}{2} times \frac{2\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{10}}{2\times 5}\sqrt{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{10}}{5}\sqrt{2}$$

Express \frac{\sqrt{10}}{5}\sqrt{2} as a single fraction.

$$\frac{\sqrt{10}\sqrt{2}}{5}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{\sqrt{2}\sqrt{5}\sqrt{2}}{5}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{2\sqrt{5}}{5}$$