Solve sqrt{90}-2sqrt{40}+5sqrt{4/5}

$$\sqrt { 90 } - 2 \sqrt { 40 } + 5 \sqrt { \frac { 4 } { 5 } }$$

Evaluate

$$2\sqrt{5}-\sqrt{10}\approx 1.309858295$$

Solution Steps

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$3\sqrt{10}-2\sqrt{40}+5\sqrt{\frac{4}{5}}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

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

Multiply -2 and 2 to get -4.

$$3\sqrt{10}-4\sqrt{10}+5\sqrt{\frac{4}{5}}$$

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

$$-\sqrt{10}+5\sqrt{\frac{4}{5}}$$

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

$$-\sqrt{10}+5\times \frac{\sqrt{4}}{\sqrt{5}}$$

Calculate the square root of 4 and get 2.

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

Rationalize the denominator of \frac{2}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

Cancel out 5 and 5.

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