Solve frac{sqrt{3}}{sqrt{2}}*sqrt{14/10}divsqrt{21/4}

$$\frac { \sqrt { 3 } } { \sqrt { 2 } } \times \sqrt { \frac { 14 } { 10 } } \div \sqrt { \frac { 21 } { 4 } }$$

Evaluate

$$\frac{\sqrt{10}}{5}\approx 0.632455532$$

Solution Steps

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

$$\frac{\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sqrt{\frac{14}{10}}}{\sqrt{\frac{21}{4}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{\frac{14}{10}}}{\sqrt{\frac{21}{4}}}$$

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

$$\frac{\frac{\sqrt{6}}{2}\sqrt{\frac{14}{10}}}{\sqrt{\frac{21}{4}}}$$

Reduce the fraction \frac{14}{10} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{\sqrt{6}}{2}\sqrt{\frac{7}{5}}}{\sqrt{\frac{21}{4}}}$$

Rewrite the square root of the division \sqrt{\frac{7}{5}} as the division of square roots \frac{\sqrt{7}}{\sqrt{5}}.

$$\frac{\frac{\sqrt{6}}{2}\times \frac{\sqrt{7}}{\sqrt{5}}}{\sqrt{\frac{21}{4}}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{\sqrt{6}}{2}\times \frac{\sqrt{7}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\sqrt{\frac{21}{4}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{\sqrt{6}}{2}\times \frac{\sqrt{7}\sqrt{5}}{5}}{\sqrt{\frac{21}{4}}}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{6}}{2}\times \frac{\sqrt{35}}{5}}{\sqrt{\frac{21}{4}}}$$

Multiply \frac{\sqrt{6}}{2} times \frac{\sqrt{35}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\sqrt{6}\sqrt{35}}{2\times 5}}{\sqrt{\frac{21}{4}}}$$

Rewrite the square root of the division \sqrt{\frac{21}{4}} as the division of square roots \frac{\sqrt{21}}{\sqrt{4}}.

$$\frac{\frac{\sqrt{6}\sqrt{35}}{2\times 5}}{\frac{\sqrt{21}}{\sqrt{4}}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{\sqrt{6}\sqrt{35}}{2\times 5}}{\frac{\sqrt{21}}{2}}$$

Divide \frac{\sqrt{6}\sqrt{35}}{2\times 5} by \frac{\sqrt{21}}{2} by multiplying \frac{\sqrt{6}\sqrt{35}}{2\times 5} by the reciprocal of \frac{\sqrt{21}}{2}.

$$\frac{\sqrt{6}\sqrt{35}\times 2}{2\times 5\sqrt{21}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{6}\sqrt{35}}{5\sqrt{21}}$$

Rationalize the denominator of \frac{\sqrt{6}\sqrt{35}}{5\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{\sqrt{6}\sqrt{35}\sqrt{21}}{5\left(\sqrt{21}\right)^{2}}$$

The square of \sqrt{21} is 21.

$$\frac{\sqrt{6}\sqrt{35}\sqrt{21}}{5\times 21}$$

To multiply \sqrt{6} and \sqrt{35}, multiply the numbers under the square root.

$$\frac{\sqrt{210}\sqrt{21}}{5\times 21}$$

Factor 210=21\times 10. Rewrite the square root of the product \sqrt{21\times 10} as the product of square roots \sqrt{21}\sqrt{10}.

$$\frac{\sqrt{21}\sqrt{10}\sqrt{21}}{5\times 21}$$

Multiply \sqrt{21} and \sqrt{21} to get 21.

$$\frac{21\sqrt{10}}{5\times 21}$$

Multiply 5 and 21 to get 105.

$$\frac{21\sqrt{10}}{105}$$

Divide 21\sqrt{10} by 105 to get \frac{1}{5}\sqrt{10}.

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