Solve 2sqrt{2/3}*(3sqrt{2/3}-4sqrt{3/2})

$$2 \sqrt { \frac { 2 } { 3 } } \times ( 3 \sqrt { \frac { 2 } { 3 } } - 4 \sqrt { \frac { 3 } { 2 } } )$$

Evaluate

$$-4$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3 and 3.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Cancel out 2, the greatest common factor in 4 and 2.

$$2\times \frac{\sqrt{6}}{3}\left(\sqrt{6}-2\sqrt{6}\right)$$

Combine \sqrt{6} and -2\sqrt{6} to get -\sqrt{6}.

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

Multiply 2 and -1 to get -2.

$$-2\times \frac{\sqrt{6}}{3}\sqrt{6}$$

Express -2\times \frac{\sqrt{6}}{3} as a single fraction.

$$\frac{-2\sqrt{6}}{3}\sqrt{6}$$

Express \frac{-2\sqrt{6}}{3}\sqrt{6} as a single fraction.

$$\frac{-2\sqrt{6}\sqrt{6}}{3}$$

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

$$\frac{-2\times 6}{3}$$

Multiply -2 and 6 to get -12.

$$\frac{-12}{3}$$

Divide -12 by 3 to get -4.

$$-4$$

Factor

$$-4$$