Solve sqrt{5/3}divsqrt{7/3}*sqrt{7/5}

$$\sqrt{ \frac{ 5 }{ 3 } } \div \sqrt{ \frac{ 7 }{ 3 } } \times \sqrt{ \frac{ 7 }{ 5 } }$$

Evaluate

$$1$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{5}}{\sqrt{3}}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{7}{5}}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{7}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{5}\sqrt{3}}{3}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{7}{5}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{15}}{3}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{7}{5}}$$

Rewrite the square root of the division \sqrt{\frac{7}{3}} as the division of square roots \frac{\sqrt{7}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{7}}{\sqrt{3}}}\sqrt{\frac{7}{5}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\sqrt{\frac{7}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{7}\sqrt{3}}{3}}\sqrt{\frac{7}{5}}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{21}}{3}}\sqrt{\frac{7}{5}}$$

Divide \frac{\sqrt{15}}{3} by \frac{\sqrt{21}}{3} by multiplying \frac{\sqrt{15}}{3} by the reciprocal of \frac{\sqrt{21}}{3}.

$$\frac{\sqrt{15}\times 3}{3\sqrt{21}}\sqrt{\frac{7}{5}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{15}}{\sqrt{21}}\sqrt{\frac{7}{5}}$$

Rationalize the denominator of \frac{\sqrt{15}}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{\sqrt{15}\sqrt{21}}{\left(\sqrt{21}\right)^{2}}\sqrt{\frac{7}{5}}$$

The square of \sqrt{21} is 21.

$$\frac{\sqrt{15}\sqrt{21}}{21}\sqrt{\frac{7}{5}}$$

To multiply \sqrt{15} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{\sqrt{315}}{21}\sqrt{\frac{7}{5}}$$

Factor 315=3^{2}\times 35. Rewrite the square root of the product \sqrt{3^{2}\times 35} as the product of square roots \sqrt{3^{2}}\sqrt{35}. Take the square root of 3^{2}.

$$\frac{3\sqrt{35}}{21}\sqrt{\frac{7}{5}}$$

Divide 3\sqrt{35} by 21 to get \frac{1}{7}\sqrt{35}.

$$\frac{1}{7}\sqrt{35}\sqrt{\frac{7}{5}}$$

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

$$\frac{1}{7}\sqrt{35}\times \frac{\sqrt{7}}{\sqrt{5}}$$

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

$$\frac{1}{7}\sqrt{35}\times \frac{\sqrt{7}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{1}{7}\sqrt{35}\times \frac{\sqrt{7}\sqrt{5}}{5}$$

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

$$\frac{1}{7}\sqrt{35}\times \frac{\sqrt{35}}{5}$$

Multiply \frac{1}{7} times \frac{\sqrt{35}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{35}}{7\times 5}\sqrt{35}$$

Multiply 7 and 5 to get 35.

$$\frac{\sqrt{35}}{35}\sqrt{35}$$

Express \frac{\sqrt{35}}{35}\sqrt{35} as a single fraction.

$$\frac{\sqrt{35}\sqrt{35}}{35}$$

Multiply \sqrt{35} and \sqrt{35} to get 35.

$$\frac{35}{35}$$

Divide 35 by 35 to get 1.

$$1$$

Factor

$$1$$