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

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

Evaluate

$$10\sqrt{2}+16-6\sqrt{6}-8\sqrt{3}\approx 1.588790706$$

Solution Steps

Square \sqrt{6}-\sqrt{2}+\sqrt{3}-2.

$$2\sqrt{3}\sqrt{6}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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}.

$$2\sqrt{3}\sqrt{3}\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$2\times 3\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Multiply 2 and 3 to get 6.

$$6\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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}.

$$6\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{2}\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-2\sqrt{6}-2\times 2\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Multiply -2 and 2 to get -4.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{2} is 2.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{3} is 3.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+2+3+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Add 2 and 3 to get 5.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+5+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{6} is 6.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+5+6-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Add 5 and 6 to get 11.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+11-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Combine -4\sqrt{3} and -4\sqrt{3} to get -8\sqrt{3}.

$$6\sqrt{2}-2\sqrt{6}-8\sqrt{3}+11-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-6\sqrt{6}-8\sqrt{3}+11+4\sqrt{2}+4+1$$

Combine 6\sqrt{2} and 4\sqrt{2} to get 10\sqrt{2}.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+11+4+1$$

Add 11 and 4 to get 15.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+15+1$$

Add 15 and 1 to get 16.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+16$$

Expand

$$10 \sqrt{2} + 16 - 6 \sqrt{6} - 8 \sqrt{3} = 1.588790706$$

Solution Steps

Square \sqrt{6}-\sqrt{2}+\sqrt{3}-2.

$$2\sqrt{3}\sqrt{6}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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}.

$$2\sqrt{3}\sqrt{3}\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$2\times 3\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Multiply 2 and 3 to get 6.

$$6\sqrt{2}-2\sqrt{2}\sqrt{3}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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}.

$$6\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{2}\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-2\sqrt{6}-2\times 2\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Multiply -2 and 2 to get -4.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{2} is 2.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{3} is 3.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+2+3+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Add 2 and 3 to get 5.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+5+\left(\sqrt{6}\right)^{2}-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

The square of \sqrt{6} is 6.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+5+6-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Add 5 and 6 to get 11.

$$6\sqrt{2}-2\sqrt{6}-4\sqrt{3}+11-4\sqrt{3}-4\sqrt{6}+4\sqrt{2}+4+1$$

Combine -4\sqrt{3} and -4\sqrt{3} to get -8\sqrt{3}.

$$6\sqrt{2}-2\sqrt{6}-8\sqrt{3}+11-4\sqrt{6}+4\sqrt{2}+4+1$$

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

$$6\sqrt{2}-6\sqrt{6}-8\sqrt{3}+11+4\sqrt{2}+4+1$$

Combine 6\sqrt{2} and 4\sqrt{2} to get 10\sqrt{2}.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+11+4+1$$

Add 11 and 4 to get 15.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+15+1$$

Add 15 and 1 to get 16.

$$10\sqrt{2}-6\sqrt{6}-8\sqrt{3}+16$$