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

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

Evaluate

$$-2\sqrt{6}-4\approx -8.898979486$$

Solution Steps

Apply the distributive property by multiplying each term of 1+\sqrt{2}+\sqrt{3} by each term of 1-\sqrt{2}-\sqrt{3}.

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

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

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

The square of \sqrt{2} is 2.

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

Subtract 2 from 1 to get -1.

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

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

$$-1-2\sqrt{6}-3$$

Subtract 3 from -1 to get -4.

$$-4-2\sqrt{6}$$