Solve 2sqrt{2/3}-3sqrt{3/2}+sqrt{24}

$$2 \sqrt { \frac { 2 } { 3 } } - 3 \sqrt { \frac { 3 } { 2 } } + \sqrt { 24 }$$

Evaluate

$$\frac{7\sqrt{6}}{6}\approx 2.857738033$$

Solution Steps

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

$$2\times \frac{\sqrt{2}}{\sqrt{3}}-3\sqrt{\frac{3}{2}}+\sqrt{24}$$

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

$$2\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-3\sqrt{\frac{3}{2}}+\sqrt{24}$$

The square of \sqrt{3} is 3.

$$2\times \frac{\sqrt{2}\sqrt{3}}{3}-3\sqrt{\frac{3}{2}}+\sqrt{24}$$

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

$$2\times \frac{\sqrt{6}}{3}-3\sqrt{\frac{3}{2}}+\sqrt{24}$$

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

$$\frac{2\sqrt{6}}{3}-3\sqrt{\frac{3}{2}}+\sqrt{24}$$

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

$$\frac{2\sqrt{6}}{3}-3\times \frac{\sqrt{3}}{\sqrt{2}}+\sqrt{24}$$

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

$$\frac{2\sqrt{6}}{3}-3\times \frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{24}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{6}}{3}-3\times \frac{\sqrt{3}\sqrt{2}}{2}+\sqrt{24}$$

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

$$\frac{2\sqrt{6}}{3}-3\times \frac{\sqrt{6}}{2}+\sqrt{24}$$

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

$$\frac{2\sqrt{6}}{3}+\frac{-3\sqrt{6}}{2}+\sqrt{24}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

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

Combine \frac{2\sqrt{6}}{3} and 2\sqrt{6} to get \frac{8}{3}\sqrt{6}.

$$\frac{8}{3}\sqrt{6}+\frac{-3\sqrt{6}}{2}$$

Combine \frac{8}{3}\sqrt{6} and \frac{-3\sqrt{6}}{2} to get \frac{7}{6}\sqrt{6}.

$$\frac{7}{6}\sqrt{6}$$