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

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

Evaluate

$$\sqrt{3}+5\sqrt{2}+11-\sqrt{6}\approx 17.353628877$$

Solution Steps

Square \sqrt{6}+\sqrt{3}+\sqrt{2}-1.

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

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

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{6} is 6.

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

Add 5 and 6 to get 11.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+11-2\sqrt{2}-2\sqrt{3}-2\sqrt{6}+1-\left(\sqrt{6}+\sqrt{3}-\sqrt{2}+1\right)$$

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

$$2\sqrt{6}+4\sqrt{3}+4\sqrt{2}+11-2\sqrt{3}-2\sqrt{6}+1-\left(\sqrt{6}+\sqrt{3}-\sqrt{2}+1\right)$$

Combine 4\sqrt{3} and -2\sqrt{3} to get 2\sqrt{3}.

$$2\sqrt{6}+2\sqrt{3}+4\sqrt{2}+11-2\sqrt{6}+1-\left(\sqrt{6}+\sqrt{3}-\sqrt{2}+1\right)$$

Combine 2\sqrt{6} and -2\sqrt{6} to get 0.

$$2\sqrt{3}+4\sqrt{2}+11+1-\left(\sqrt{6}+\sqrt{3}-\sqrt{2}+1\right)$$

Add 11 and 1 to get 12.

$$2\sqrt{3}+4\sqrt{2}+12-\left(\sqrt{6}+\sqrt{3}-\sqrt{2}+1\right)$$

To find the opposite of \sqrt{6}+\sqrt{3}-\sqrt{2}+1, find the opposite of each term.

$$2\sqrt{3}+4\sqrt{2}+12-\sqrt{6}-\sqrt{3}+\sqrt{2}-1$$

Combine 2\sqrt{3} and -\sqrt{3} to get \sqrt{3}.

$$\sqrt{3}+4\sqrt{2}+12-\sqrt{6}+\sqrt{2}-1$$

Combine 4\sqrt{2} and \sqrt{2} to get 5\sqrt{2}.

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

Subtract 1 from 12 to get 11.

$$\sqrt{3}+5\sqrt{2}+11-\sqrt{6}$$