Solve 3sqrt{2/3}div1/2sqrt{2/7}*(-1/8sqrt{4})

$$3 \sqrt { \frac { 2 } { 3 } } \div \frac { 1 } { 2 } \sqrt { \frac { 2 } { 7 } } \times ( - \frac { 1 } { 8 } \sqrt { 4 } )$$

Evaluate

$$-\frac{\sqrt{21}}{7}\approx -0.654653671$$

Solution Steps

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

$$\frac{3\times \frac{\sqrt{2}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

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

$$\frac{3\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

The square of \sqrt{3} is 3.

$$\frac{3\times \frac{\sqrt{2}\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

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

$$\frac{3\times \frac{\sqrt{6}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

Cancel out 3 and 3.

$$\frac{\sqrt{6}}{\frac{1}{2}}\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

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

$$\sqrt{6}\times 2\sqrt{\frac{2}{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

Rewrite the square root of the division \sqrt{\frac{2}{7}} as the division of square roots \frac{\sqrt{2}}{\sqrt{7}}.

$$\sqrt{6}\times 2\times \frac{\sqrt{2}}{\sqrt{7}}\left(-\frac{1}{8}\right)\sqrt{4}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\sqrt{6}\times 2\times \frac{\sqrt{2}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}\left(-\frac{1}{8}\right)\sqrt{4}$$

The square of \sqrt{7} is 7.

$$\sqrt{6}\times 2\times \frac{\sqrt{2}\sqrt{7}}{7}\left(-\frac{1}{8}\right)\sqrt{4}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\sqrt{6}\times 2\times \frac{\sqrt{14}}{7}\left(-\frac{1}{8}\right)\sqrt{4}$$

Express 2\left(-\frac{1}{8}\right) as a single fraction.

$$\sqrt{6}\times \frac{2\left(-1\right)}{8}\times \frac{\sqrt{14}}{7}\sqrt{4}$$

Multiply 2 and -1 to get -2.

$$\sqrt{6}\times \frac{-2}{8}\times \frac{\sqrt{14}}{7}\sqrt{4}$$

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

$$\sqrt{6}\left(-\frac{1}{4}\right)\times \frac{\sqrt{14}}{7}\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$\sqrt{6}\left(-\frac{1}{4}\right)\times \frac{\sqrt{14}}{7}\times 2$$

Express -\frac{1}{4}\times 2 as a single fraction.

$$\sqrt{6}\times \frac{-2}{4}\times \frac{\sqrt{14}}{7}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{6}\left(-\frac{1}{2}\right)\times \frac{\sqrt{14}}{7}$$

Express \sqrt{6}\times \frac{\sqrt{14}}{7} as a single fraction.

$$\frac{\sqrt{6}\sqrt{14}}{7}\left(-\frac{1}{2}\right)$$

To multiply \sqrt{6} and \sqrt{14}, multiply the numbers under the square root.

$$\frac{\sqrt{84}}{7}\left(-\frac{1}{2}\right)$$

Factor 84=2^{2}\times 21. Rewrite the square root of the product \sqrt{2^{2}\times 21} as the product of square roots \sqrt{2^{2}}\sqrt{21}. Take the square root of 2^{2}.

$$\frac{2\sqrt{21}}{7}\left(-\frac{1}{2}\right)$$

Multiply \frac{2\sqrt{21}}{7} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2\sqrt{21}}{7\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-\sqrt{21}}{7}$$