Solve (sqrt{3}-2sqrt{12}-sqrt{6})*2sqrt{3}+5sqrt{2}

$$( \sqrt { 3 } - 2 \sqrt { 12 } - \sqrt { 6 } ) \cdot 2 \sqrt { 3 } + 5 \sqrt { 2 }$$

Evaluate

$$-\sqrt{2}-18\approx -19.414213562$$

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}.

$$2\left(\sqrt{3}-2\times 2\sqrt{3}-\sqrt{6}\right)\sqrt{3}+5\sqrt{2}$$

Multiply -2 and 2 to get -4.

$$2\left(\sqrt{3}-4\sqrt{3}-\sqrt{6}\right)\sqrt{3}+5\sqrt{2}$$

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

$$2\left(-3\sqrt{3}-\sqrt{6}\right)\sqrt{3}+5\sqrt{2}$$

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

$$\left(-6\sqrt{3}-2\sqrt{6}\right)\sqrt{3}+5\sqrt{2}$$

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

$$-6\left(\sqrt{3}\right)^{2}-2\sqrt{6}\sqrt{3}+5\sqrt{2}$$

The square of \sqrt{3} is 3.

$$-6\times 3-2\sqrt{6}\sqrt{3}+5\sqrt{2}$$

Multiply -6 and 3 to get -18.

$$-18-2\sqrt{6}\sqrt{3}+5\sqrt{2}$$

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}.

$$-18-2\sqrt{3}\sqrt{2}\sqrt{3}+5\sqrt{2}$$

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

$$-18-2\times 3\sqrt{2}+5\sqrt{2}$$

Multiply -2 and 3 to get -6.

$$-18-6\sqrt{2}+5\sqrt{2}$$

Combine -6\sqrt{2} and 5\sqrt{2} to get -\sqrt{2}.

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

Factor

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