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

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

Evaluate

$$6\sqrt{2}-7\approx 1.485281374$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{2}+\sqrt{3}-\sqrt{6} by each term of \sqrt{2}-\sqrt{3}+\sqrt{6}.

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

The square of \sqrt{2} is 2.

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

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

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

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}.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 2 to get -1.

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

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}.

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

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

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

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}.

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

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

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

Combine 2\sqrt{3} and -2\sqrt{3} to get 0.

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

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}.

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

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

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

Combine 3\sqrt{2} and 3\sqrt{2} to get 6\sqrt{2}.

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

The square of \sqrt{6} is 6.

$$-1+6\sqrt{2}-6$$

Subtract 6 from -1 to get -7.

$$-7+6\sqrt{2}$$