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

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

Evaluate

$$2$$

Solution Steps

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

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

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

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

Apply the distributive property by multiplying each term of 1+\sqrt{2}+\sqrt{3}+\sqrt{6} by each term of 1-\sqrt{3}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 1 to get -2.

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

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

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

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}.

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

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

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

$$-2+2\times 2$$

Multiply 2 and 2 to get 4.

$$-2+4$$

Add -2 and 4 to get 2.

$$2$$

Factor

$$2$$