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

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

Evaluate

$$\frac{3\sqrt{6}}{4}\approx 1.837117307$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3 and 3.

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

Divide \frac{1}{8}\sqrt{15} by \frac{1}{2} by multiplying \frac{1}{8}\sqrt{15} by the reciprocal of \frac{1}{2}.

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

Multiply \frac{1}{8} and 2 to get \frac{2}{8}.

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

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

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

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

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

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

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

The square of \sqrt{5} is 5.

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

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

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

Multiply \frac{1}{4} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 4 and 5 to get 20.

$$\sqrt{6}-\frac{\sqrt{10}}{20}\sqrt{15}$$

Express \frac{\sqrt{10}}{20}\sqrt{15} as a single fraction.

$$\sqrt{6}-\frac{\sqrt{10}\sqrt{15}}{20}$$

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\sqrt{6}-\frac{\sqrt{150}}{20}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\sqrt{6}-\frac{5\sqrt{6}}{20}$$

Divide 5\sqrt{6} by 20 to get \frac{1}{4}\sqrt{6}.

$$\sqrt{6}-\frac{1}{4}\sqrt{6}$$

Combine \sqrt{6} and -\frac{1}{4}\sqrt{6} to get \frac{3}{4}\sqrt{6}.

$$\frac{3}{4}\sqrt{6}$$