Solve sqrt{6}(-sqrt{2}+sqrt{6}-sqrt{24}+sqrt{18})

$$\sqrt { 6 } ( - \sqrt { 2 } + \sqrt { 6 } - \sqrt { 24 } + \sqrt { 18 } )$$

Evaluate

$$4\sqrt{3}-6\approx 0.92820323$$

Solution Steps

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}.

$$\sqrt{6}\left(-\sqrt{2}+\sqrt{6}-2\sqrt{6}+\sqrt{18}\right)$$

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

$$\sqrt{6}\left(-\sqrt{2}-\sqrt{6}+\sqrt{18}\right)$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

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

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

The square of \sqrt{6} is 6.

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

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}.

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

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

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

Multiply 3 and 2 to get 6.

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

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}.

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

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

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

Multiply 2 and -1 to get -2.

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

Combine -2\sqrt{3} and 6\sqrt{3} to get 4\sqrt{3}.

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