Solve {left(sqrt{2}-sqrt{3}right)}^2-2sqrt{1/3}*3sqrt{12}

$${ \left( \sqrt{ 2 } - \sqrt{ 3 } \right) }^{ 2 } -2 \sqrt{ \frac{ 1 }{ 3 } } \times 3 \sqrt{ 12 }$$

Evaluate

$$-2\sqrt{6}-7\approx -11.898979486$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-\sqrt{3}\right)^{2}.

$$\left(\sqrt{2}\right)^{2}-2\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^{2}-2\times 3\sqrt{\frac{1}{3}}\sqrt{12}$$

The square of \sqrt{2} is 2.

$$2-2\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^{2}-2\times 3\sqrt{\frac{1}{3}}\sqrt{12}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2-2\sqrt{6}+\left(\sqrt{3}\right)^{2}-2\times 3\sqrt{\frac{1}{3}}\sqrt{12}$$

The square of \sqrt{3} is 3.

$$2-2\sqrt{6}+3-2\times 3\sqrt{\frac{1}{3}}\sqrt{12}$$

Add 2 and 3 to get 5.

$$5-2\sqrt{6}-2\times 3\sqrt{\frac{1}{3}}\sqrt{12}$$

Multiply 2 and 3 to get 6.

$$5-2\sqrt{6}-6\sqrt{\frac{1}{3}}\sqrt{12}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$5-2\sqrt{6}-6\times \frac{\sqrt{1}}{\sqrt{3}}\sqrt{12}$$

Calculate the square root of 1 and get 1.

$$5-2\sqrt{6}-6\times \frac{1}{\sqrt{3}}\sqrt{12}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$5-2\sqrt{6}-6\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{12}$$

The square of \sqrt{3} is 3.

$$5-2\sqrt{6}-6\times \frac{\sqrt{3}}{3}\sqrt{12}$$

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}.

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

Multiply 6 and 2 to get 12.

$$5-2\sqrt{6}-12\times \frac{\sqrt{3}}{3}\sqrt{3}$$

Cancel out 3, the greatest common factor in 12 and 3.

$$5-2\sqrt{6}-4\sqrt{3}\sqrt{3}$$

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

$$5-2\sqrt{6}-4\times 3$$

Multiply 4 and 3 to get 12.

$$5-2\sqrt{6}-12$$

Subtract 12 from 5 to get -7.

$$-7-2\sqrt{6}$$