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

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

Evaluate

$$\sqrt{5}+4\sqrt{30}-8\sqrt{6}\approx 4.549052335$$

Solution Steps

Apply the distributive property by multiplying each term of 3-\sqrt{5} by each term of \sqrt{10}-\sqrt{2}.

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

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}.

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

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

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

Combine -3\sqrt{2} and -5\sqrt{2} to get -8\sqrt{2}.

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

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

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

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

$$\left(4\sqrt{10}-8\sqrt{2}\right)\sqrt{3}+\sqrt{5}$$

Use the distributive property to multiply 4\sqrt{10}-8\sqrt{2} by \sqrt{3}.

$$4\sqrt{10}\sqrt{3}-8\sqrt{2}\sqrt{3}+\sqrt{5}$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$4\sqrt{30}-8\sqrt{2}\sqrt{3}+\sqrt{5}$$

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

$$4\sqrt{30}-8\sqrt{6}+\sqrt{5}$$