Solve {left(sqrt{2}-1right)}^2+2left(sqrt{3}-sqrt{2}right)left(sqrt{6}+2right)

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

Evaluate

$$3$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-1\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

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

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

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

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

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}.

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

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

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

Multiply -2 and 2 to get -4.

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

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

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

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

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

Combine -2\sqrt{2} and 2\sqrt{2} to get 0.

$$3$$

Factor

$$3$$