Solve (sqrt{1/2}-frac{sqrt{3}}{3})*sqrt{24}

$$( \sqrt { \frac { 1 } { 2 } } - \frac { \sqrt { 3 } } { 3 } ) \times \sqrt { 24 }$$

Evaluate

$$2\left(\sqrt{3}-\sqrt{2}\right)\approx 0.63567449$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$\left(\frac{\sqrt{1}}{\sqrt{2}}-\frac{\sqrt{3}}{3}\right)\sqrt{24}$$

Calculate the square root of 1 and get 1.

$$\left(\frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{3}\right)\sqrt{24}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{\sqrt{2}}{2} times \frac{3}{3}. Multiply \frac{\sqrt{3}}{3} times \frac{2}{2}.

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

Since \frac{3\sqrt{2}}{6} and \frac{2\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

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

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}.

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

Cancel out 6, the greatest common factor in 2 and 6.

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

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

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

Use the distributive property to multiply 3\sqrt{2}-2\sqrt{3} by \sqrt{6}.

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply 3 and 2 to get 6.

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

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply -2 and 3 to get -6.

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

Factor

$$2 {(\sqrt{3} - \sqrt{2})} = 0.63567449$$