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

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

Evaluate

$$-3$$

Solution Steps

Multiply 2 and 3 to get 6.

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

Add 6 and 2 to get 8.

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

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

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

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}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3 and 3.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

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

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

The square of \sqrt{5} is 5.

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

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

$$4\sqrt{6}\times \frac{\sqrt{10}}{5}\left(-\frac{1}{8}\right)\sqrt{15}$$

Express 4\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{4\left(-1\right)}{8}\sqrt{6}\times \frac{\sqrt{10}}{5}\sqrt{15}$$

Multiply 4 and -1 to get -4.

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

Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

$$-\frac{1}{2}\sqrt{6}\times \frac{\sqrt{10}}{5}\sqrt{15}$$

Multiply -\frac{1}{2} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\sqrt{10}}{2\times 5}\sqrt{6}\sqrt{15}$$

Express \frac{-\sqrt{10}}{2\times 5}\sqrt{15} as a single fraction.

$$\frac{-\sqrt{10}\sqrt{15}}{2\times 5}\sqrt{6}$$

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

$$\frac{-\sqrt{150}}{2\times 5}\sqrt{6}$$

Multiply 2 and 5 to get 10.

$$\frac{-\sqrt{150}}{10}\sqrt{6}$$

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}.

$$\frac{-5\sqrt{6}}{10}\sqrt{6}$$

Divide -5\sqrt{6} by 10 to get -\frac{1}{2}\sqrt{6}.

$$-\frac{1}{2}\sqrt{6}\sqrt{6}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$-\frac{1}{2}\times 6$$

Express -\frac{1}{2}\times 6 as a single fraction.

$$\frac{-6}{2}$$

Divide -6 by 2 to get -3.

$$-3$$

Factor

$$-3$$