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

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

Evaluate

$$4\left(\sqrt{6}-\sqrt{10}\right)\approx -2.85115167$$

Solution Steps

Square \sqrt{2}+\sqrt{3}-\sqrt{5}.

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{5} is 5.

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

Add 5 and 5 to get 10.

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

Square \sqrt{2}-\sqrt{3}+\sqrt{5}.

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{5} is 5.

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

Add 5 and 5 to get 10.

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

To find the opposite of 2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+10, find the opposite of each term.

$$2\sqrt{6}-2\sqrt{10}-2\sqrt{15}+10-2\sqrt{10}+2\sqrt{6}+2\sqrt{15}-10$$

Combine -2\sqrt{10} and -2\sqrt{10} to get -4\sqrt{10}.

$$2\sqrt{6}-4\sqrt{10}-2\sqrt{15}+10+2\sqrt{6}+2\sqrt{15}-10$$

Combine 2\sqrt{6} and 2\sqrt{6} to get 4\sqrt{6}.

$$4\sqrt{6}-4\sqrt{10}-2\sqrt{15}+10+2\sqrt{15}-10$$

Combine -2\sqrt{15} and 2\sqrt{15} to get 0.

$$4\sqrt{6}-4\sqrt{10}+10-10$$

Subtract 10 from 10 to get 0.

$$4\sqrt{6}-4\sqrt{10}$$

Expand

$$4 \sqrt{6} - 4 \sqrt{10} = -2.85115167$$

Solution Steps

Square \sqrt{2}+\sqrt{3}-\sqrt{5}.

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{5} is 5.

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

Add 5 and 5 to get 10.

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

Square \sqrt{2}-\sqrt{3}+\sqrt{5}.

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Add 2 and 3 to get 5.

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

The square of \sqrt{5} is 5.

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

Add 5 and 5 to get 10.

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

To find the opposite of 2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+10, find the opposite of each term.

$$2\sqrt{6}-2\sqrt{10}-2\sqrt{15}+10-2\sqrt{10}+2\sqrt{6}+2\sqrt{15}-10$$

Combine -2\sqrt{10} and -2\sqrt{10} to get -4\sqrt{10}.

$$2\sqrt{6}-4\sqrt{10}-2\sqrt{15}+10+2\sqrt{6}+2\sqrt{15}-10$$

Combine 2\sqrt{6} and 2\sqrt{6} to get 4\sqrt{6}.

$$4\sqrt{6}-4\sqrt{10}-2\sqrt{15}+10+2\sqrt{15}-10$$

Combine -2\sqrt{15} and 2\sqrt{15} to get 0.

$$4\sqrt{6}-4\sqrt{10}+10-10$$

Subtract 10 from 10 to get 0.

$$4\sqrt{6}-4\sqrt{10}$$