Solve (sqrt{3}+sqrt{5}-sqrt{2})(sqrt{3}-sqrt{5}+sqrt{2})

$$( \sqrt { 3 } + \sqrt { 5 } - \sqrt { 2 } ) ( \sqrt { 3 } - \sqrt { 5 } + \sqrt { 2 } )$$

Evaluate

$$2\left(\sqrt{10}-2\right)\approx 2.32455532$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{3}+\sqrt{5}-\sqrt{2} by each term of \sqrt{3}-\sqrt{5}+\sqrt{2}.

$$\left(\sqrt{3}\right)^{2}-\sqrt{3}\sqrt{5}+\sqrt{3}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$3-\sqrt{3}\sqrt{5}+\sqrt{3}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$3-\sqrt{15}+\sqrt{3}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

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

$$3-\sqrt{15}+\sqrt{6}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$3-\sqrt{15}+\sqrt{6}+\sqrt{15}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

Combine -\sqrt{15} and \sqrt{15} to get 0.

$$3+\sqrt{6}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$3+\sqrt{6}-5+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

Subtract 5 from 3 to get -2.

$$-2+\sqrt{6}+\sqrt{5}\sqrt{2}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$-2+\sqrt{6}+\sqrt{10}-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

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

$$-2+\sqrt{6}+\sqrt{10}-\sqrt{6}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

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

$$-2+\sqrt{10}+\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$-2+\sqrt{10}+\sqrt{10}-\left(\sqrt{2}\right)^{2}$$

Combine \sqrt{10} and \sqrt{10} to get 2\sqrt{10}.

$$-2+2\sqrt{10}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

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

Subtract 2 from -2 to get -4.

$$-4+2\sqrt{10}$$