Solve (sqrt{3}-sqrt{1/2})sqrt{6}

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

Evaluate

$$\sqrt{3}\left(\sqrt{6}-1\right)\approx 2.51058988$$

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}}.

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{2}{2}.

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

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

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

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

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

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

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

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}.

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

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

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

Divide each term of 6\sqrt{2}-2\sqrt{3} by 2 to get 3\sqrt{2}-\sqrt{3}.

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