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

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

Evaluate

$$\sqrt{3}-6\sqrt{2}\approx -6.753230567$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

Since \frac{\sqrt{2}}{2} and \frac{2\left(-2\right)\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{2}+2\left(-2\right)\sqrt{3}.

$$\sqrt{6}\times \frac{\sqrt{2}-4\sqrt{3}}{2}$$

Express \sqrt{6}\times \frac{\sqrt{2}-4\sqrt{3}}{2} as a single fraction.

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

Use the distributive property to multiply \sqrt{6} by \sqrt{2}-4\sqrt{3}.

$$\frac{\sqrt{6}\sqrt{2}-4\sqrt{6}\sqrt{3}}{2}$$

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}.

$$\frac{\sqrt{2}\sqrt{3}\sqrt{2}-4\sqrt{6}\sqrt{3}}{2}$$

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

$$\frac{2\sqrt{3}-4\sqrt{6}\sqrt{3}}{2}$$

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}.

$$\frac{2\sqrt{3}-4\sqrt{3}\sqrt{2}\sqrt{3}}{2}$$

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

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

Multiply -4 and 3 to get -12.

$$\frac{2\sqrt{3}-12\sqrt{2}}{2}$$

Divide each term of 2\sqrt{3}-12\sqrt{2} by 2 to get \sqrt{3}-6\sqrt{2}.

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