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

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

Evaluate

$$6$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{6}\right)^{2}.

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

The square of \sqrt{6} is 6.

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

Add 9 and 6 to get 15.

$$\left(15-6\sqrt{6}\right)\left(\sqrt{6}+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{6}+2\right)^{2}.

$$\left(15-6\sqrt{6}\right)\left(\left(\sqrt{6}\right)^{2}+4\sqrt{6}+4\right)$$

The square of \sqrt{6} is 6.

$$\left(15-6\sqrt{6}\right)\left(6+4\sqrt{6}+4\right)$$

Add 6 and 4 to get 10.

$$\left(15-6\sqrt{6}\right)\left(10+4\sqrt{6}\right)$$

Use the distributive property to multiply 15-6\sqrt{6} by 10+4\sqrt{6} and combine like terms.

$$150-24\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$150-24\times 6$$

Multiply -24 and 6 to get -144.

$$150-144$$

Subtract 144 from 150 to get 6.

$$6$$

Factor

$$2\times 3$$