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

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

Evaluate

$$-2\sqrt{2}\approx -2.828427125$$

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 -2\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 3 to get 0.

$$-2\sqrt{2}$$