Solve left(sqrt{5}-2sqrt{2}right)left(sqrt{20}+sqrt{2}right)

$$\left( \sqrt{ 5 } -2 \sqrt{ 2 } \right) \left( \sqrt{ 20 } + \sqrt{ 2 } \right)$$

Evaluate

$$6-3\sqrt{10}\approx -3.486832981$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

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

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

$$2\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-4\sqrt{2}\sqrt{5}-2\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$2\times 5+\sqrt{5}\sqrt{2}-4\sqrt{2}\sqrt{5}-2\left(\sqrt{2}\right)^{2}$$

Multiply 2 and 5 to get 10.

$$10+\sqrt{5}\sqrt{2}-4\sqrt{2}\sqrt{5}-2\left(\sqrt{2}\right)^{2}$$

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

$$10+\sqrt{10}-4\sqrt{2}\sqrt{5}-2\left(\sqrt{2}\right)^{2}$$

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

$$10+\sqrt{10}-4\sqrt{10}-2\left(\sqrt{2}\right)^{2}$$

Combine \sqrt{10} and -4\sqrt{10} to get -3\sqrt{10}.

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

The square of \sqrt{2} is 2.

$$10-3\sqrt{10}-2\times 2$$

Multiply -2 and 2 to get -4.

$$10-3\sqrt{10}-4$$

Subtract 4 from 10 to get 6.

$$6-3\sqrt{10}$$