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

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

Evaluate

$$4\sqrt{3}+11-2\sqrt{6}-6\sqrt{2}\approx 4.54394237$$

Solution Steps

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

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

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}.

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

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

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

Multiply -2 and 3 to get -6.

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{6} is 6.

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

Add 5 and 6 to get 11.

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

Expand

$$4 \sqrt{3} + 11 - 2 \sqrt{6} - 6 \sqrt{2} = 4.54394237$$

Solution Steps

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

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

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}.

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

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

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

Multiply -2 and 3 to get -6.

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{6} is 6.

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

Add 5 and 6 to get 11.

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