Solve 3sqrt{8/3}*(-1/8sqrt{15})div1/2sqrt{2/5}

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

Evaluate

$$-3$$

Solution Steps

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

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

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}.

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

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

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

The square of \sqrt{3} is 3.

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

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

$$\frac{3\times \frac{2\sqrt{6}}{3}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Express 3\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{\frac{3\left(-1\right)}{8}\times \frac{2\sqrt{6}}{3}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Multiply 3 and -1 to get -3.

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

Fraction \frac{-3}{8} can be rewritten as -\frac{3}{8} by extracting the negative sign.

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

Multiply -\frac{3}{8} times \frac{2\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Express \frac{-\sqrt{6}}{4}\sqrt{15} as a single fraction.

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

Divide \frac{-\sqrt{6}\sqrt{15}}{4} by \frac{1}{2} by multiplying \frac{-\sqrt{6}\sqrt{15}}{4} by the reciprocal of \frac{1}{2}.

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

Multiply -1 and 2 to get -2.

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

To multiply \sqrt{6} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{-2\sqrt{90}}{4}\sqrt{\frac{2}{5}}$$

Divide -2\sqrt{90} by 4 to get -\frac{1}{2}\sqrt{90}.

$$-\frac{1}{2}\sqrt{90}\sqrt{\frac{2}{5}}$$

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$-\frac{1}{2}\times 3\sqrt{10}\sqrt{\frac{2}{5}}$$

Express -\frac{1}{2}\times 3 as a single fraction.

$$\frac{-3}{2}\sqrt{10}\sqrt{\frac{2}{5}}$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

$$-\frac{3}{2}\sqrt{10}\sqrt{\frac{2}{5}}$$

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

$$-\frac{3}{2}\sqrt{10}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

$$-\frac{3}{2}\sqrt{10}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$-\frac{3}{2}\sqrt{10}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

$$-\frac{3}{2}\sqrt{10}\times \frac{\sqrt{10}}{5}$$

Multiply -\frac{3}{2} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-3\sqrt{10}}{2\times 5}\sqrt{10}$$

Multiply 2 and 5 to get 10.

$$\frac{-3\sqrt{10}}{10}\sqrt{10}$$

Express \frac{-3\sqrt{10}}{10}\sqrt{10} as a single fraction.

$$\frac{-3\sqrt{10}\sqrt{10}}{10}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{-3\times 10}{10}$$

Cancel out 10 and 10.

$$-3$$

Factor

$$-3$$