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

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

Evaluate

$$2\sqrt{6}\approx 4.898979486$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

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

$$5+2\sqrt{6}+\sqrt{10}+\sqrt{15}-\sqrt{5}\sqrt{2}-\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

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

$$5+2\sqrt{6}+\sqrt{10}+\sqrt{15}-\sqrt{10}-\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

Combine \sqrt{10} and -\sqrt{10} to get 0.

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

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

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

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

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

The square of \sqrt{5} is 5.

$$5+2\sqrt{6}-5$$

Subtract 5 from 5 to get 0.

$$2\sqrt{6}$$