Solve (sqrt{3}-sqrt{6}+sqrt{12}-sqrt{24})*frac{sqrt{6}}{2}

$$( \sqrt { 3 } - \sqrt { 6 } + \sqrt { 12 } - \sqrt { 24 } ) \cdot \frac { \sqrt { 6 } } { 2 }$$

Evaluate

$$\frac{9\left(\sqrt{2}-2\right)}{2}\approx -2.636038969$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Combine \sqrt{3} and 2\sqrt{3} to get 3\sqrt{3}.

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

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}.

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

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

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

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

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

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

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

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}.

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{2}-3\left(\sqrt{6}\right)^{2}}{2}$$

The square of \sqrt{6} is 6.

$$\frac{9\sqrt{2}-3\times 6}{2}$$

Multiply -3 and 6 to get -18.

$$\frac{9\sqrt{2}-18}{2}$$