Solve 3sqrt{22/3}div1/2sqrt{2/5}-1/8sqrt{15}

$$3 \sqrt{ 2 \frac{ 2 }{ 3 } } \div \frac{ 1 }{ 2 } \sqrt{ \frac{ 2 }{ 5 } } - \frac{ 1 }{ 8 } \sqrt{ 15 }$$

Evaluate

$$\frac{59\sqrt{15}}{40}\approx 5.712650436$$

Solution Steps

Multiply 2 and 3 to get 6.

$$\frac{3\sqrt{\frac{6+2}{3}}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

Add 6 and 2 to get 8.

$$\frac{3\sqrt{\frac{8}{3}}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

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

$$\frac{3\times \frac{\sqrt{8}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

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}.

$$\frac{3\times \frac{2\sqrt{2}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

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

$$\frac{3\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

The square of \sqrt{3} is 3.

$$\frac{3\times \frac{2\sqrt{2}\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

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

$$\frac{3\times \frac{2\sqrt{6}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

Cancel out 3 and 3.

$$\frac{2\sqrt{6}}{\frac{1}{2}}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

Divide 2\sqrt{6} by \frac{1}{2} by multiplying 2\sqrt{6} by the reciprocal of \frac{1}{2}.

$$2\sqrt{6}\times 2\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{6}\sqrt{\frac{2}{5}}-\frac{1}{8}\sqrt{15}$$

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

$$4\sqrt{6}\times \frac{\sqrt{2}}{\sqrt{5}}-\frac{1}{8}\sqrt{15}$$

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

$$4\sqrt{6}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{1}{8}\sqrt{15}$$

The square of \sqrt{5} is 5.

$$4\sqrt{6}\times \frac{\sqrt{2}\sqrt{5}}{5}-\frac{1}{8}\sqrt{15}$$

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

$$4\sqrt{6}\times \frac{\sqrt{10}}{5}-\frac{1}{8}\sqrt{15}$$

Express 4\times \frac{\sqrt{10}}{5} as a single fraction.

$$\frac{4\sqrt{10}}{5}\sqrt{6}-\frac{1}{8}\sqrt{15}$$

Express \frac{4\sqrt{10}}{5}\sqrt{6} as a single fraction.

$$\frac{4\sqrt{10}\sqrt{6}}{5}-\frac{1}{8}\sqrt{15}$$

To multiply \sqrt{10} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{4\sqrt{60}}{5}-\frac{1}{8}\sqrt{15}$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{4\times 2\sqrt{15}}{5}-\frac{1}{8}\sqrt{15}$$

Multiply 4 and 2 to get 8.

$$\frac{8\sqrt{15}}{5}-\frac{1}{8}\sqrt{15}$$

Combine \frac{8\sqrt{15}}{5} and -\frac{1}{8}\sqrt{15} to get \frac{59}{40}\sqrt{15}.

$$\frac{59}{40}\sqrt{15}$$