Solve 2frac{{left(2sqrt{3}-3sqrt{2}right)}^2}{6}left(5+2sqrt{6}right)

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

Evaluate

$$2$$

Solution Steps

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

$$2\times \frac{4\left(\sqrt{3}\right)^{2}-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}}{6}\left(5+2\sqrt{6}\right)$$

The square of \sqrt{3} is 3.

$$2\times \frac{4\times 3-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}}{6}\left(5+2\sqrt{6}\right)$$

Multiply 4 and 3 to get 12.

$$2\times \frac{12-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}}{6}\left(5+2\sqrt{6}\right)$$

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

$$2\times \frac{12-12\sqrt{6}+9\left(\sqrt{2}\right)^{2}}{6}\left(5+2\sqrt{6}\right)$$

The square of \sqrt{2} is 2.

$$2\times \frac{12-12\sqrt{6}+9\times 2}{6}\left(5+2\sqrt{6}\right)$$

Multiply 9 and 2 to get 18.

$$2\times \frac{12-12\sqrt{6}+18}{6}\left(5+2\sqrt{6}\right)$$

Add 12 and 18 to get 30.

$$2\times \frac{30-12\sqrt{6}}{6}\left(5+2\sqrt{6}\right)$$

Divide each term of 30-12\sqrt{6} by 6 to get 5-2\sqrt{6}.

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

Use the distributive property to multiply 2 by 5-2\sqrt{6}.

$$\left(10-4\sqrt{6}\right)\left(5+2\sqrt{6}\right)$$

Use the distributive property to multiply 10-4\sqrt{6} by 5+2\sqrt{6} and combine like terms.

$$50-8\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$50-8\times 6$$

Multiply -8 and 6 to get -48.

$$50-48$$

Subtract 48 from 50 to get 2.

$$2$$

Factor

$$2$$