Solve left(sqrt{10}-sqrt{2}right)left(sqrt{5}+sqrt{2}right)+{left(sqrt{5}-sqrt{2}right)}^2-5frac{sqrt{6}}{sqrt{3}}

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

Evaluate

$$\frac{6\sqrt{5}}{3}+5-3\sqrt{10}\approx -0.014697026$$

Solution Steps

Use the distributive property to multiply \sqrt{10}-\sqrt{2} by \sqrt{5}+\sqrt{2}.

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

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

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

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

The square of \sqrt{2} is 2.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{2}\right)^{2}.

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

The square of \sqrt{5} is 5.

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

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

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

The square of \sqrt{2} is 2.

$$5\sqrt{2}+2\sqrt{5}-\sqrt{10}-2+5-2\sqrt{10}+2-5\times \frac{\sqrt{6}}{\sqrt{3}}$$

Add 5 and 2 to get 7.

$$5\sqrt{2}+2\sqrt{5}-\sqrt{10}-2+7-2\sqrt{10}-5\times \frac{\sqrt{6}}{\sqrt{3}}$$

Add -2 and 7 to get 5.

$$5\sqrt{2}+2\sqrt{5}-\sqrt{10}+5-2\sqrt{10}-5\times \frac{\sqrt{6}}{\sqrt{3}}$$

Combine -\sqrt{10} and -2\sqrt{10} to get -3\sqrt{10}.

$$5\sqrt{2}+2\sqrt{5}-3\sqrt{10}+5-5\times \frac{\sqrt{6}}{\sqrt{3}}$$

Rewrite the division of square roots \frac{\sqrt{6}}{\sqrt{3}} as the square root of the division \sqrt{\frac{6}{3}} and perform the division.

$$5\sqrt{2}+2\sqrt{5}-3\sqrt{10}+5-5\sqrt{2}$$

Combine 5\sqrt{2} and -5\sqrt{2} to get 0.

$$2\sqrt{5}-3\sqrt{10}+5$$