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

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

Evaluate

$$2\left(\sqrt{3}-5\right)\approx -6.535898385$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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

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}.

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

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

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

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

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

The square of \sqrt{6} is 6.

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

Multiply -2 and 6 to get -12.

$$2+2\sqrt{3}-12$$

Subtract 12 from 2 to get -10.

$$-10+2\sqrt{3}$$

Factor

$$2 {(\sqrt{3} - 5)} = -6.535898385$$