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

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

Evaluate

$$6-2\sqrt{15}\approx -1.745966692$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Add 5 and 3 to get 8.

$$8-2\sqrt{15}-\sqrt{10}+\sqrt{3}\sqrt{2}+\sqrt{2}\sqrt{5}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$8-2\sqrt{15}-\sqrt{10}+\sqrt{6}+\sqrt{2}\sqrt{5}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$8-2\sqrt{15}-\sqrt{10}+\sqrt{6}+\sqrt{10}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$8-2\sqrt{15}+\sqrt{6}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

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

$$8-2\sqrt{15}+\sqrt{6}-\sqrt{6}-\left(\sqrt{2}\right)^{2}$$

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

$$8-2\sqrt{15}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$8-2\sqrt{15}-2$$

Subtract 2 from 8 to get 6.

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