Solve sqrt{6}-sqrt{20}+sqrt{2/3}+sqrt{9/5}

$$\sqrt { 6 } - \sqrt { 20 } + \sqrt { \frac { 2 } { 3 } } + \sqrt { \frac { 9 } { 5 } }$$

Evaluate

$$\frac{4\sqrt{6}}{3}-\frac{7\sqrt{5}}{5}\approx 0.135491155$$

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}.

$$\sqrt{6}-2\sqrt{5}+\sqrt{\frac{2}{3}}+\sqrt{\frac{9}{5}}$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\sqrt{6}-2\sqrt{5}+\frac{\sqrt{2}}{\sqrt{3}}+\sqrt{\frac{9}{5}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\sqrt{6}-2\sqrt{5}+\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{\frac{9}{5}}$$

The square of \sqrt{3} is 3.

$$\sqrt{6}-2\sqrt{5}+\frac{\sqrt{2}\sqrt{3}}{3}+\sqrt{\frac{9}{5}}$$

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

$$\sqrt{6}-2\sqrt{5}+\frac{\sqrt{6}}{3}+\sqrt{\frac{9}{5}}$$

Combine \sqrt{6} and \frac{\sqrt{6}}{3} to get \frac{4}{3}\sqrt{6}.

$$\frac{4}{3}\sqrt{6}-2\sqrt{5}+\sqrt{\frac{9}{5}}$$

Rewrite the square root of the division \sqrt{\frac{9}{5}} as the division of square roots \frac{\sqrt{9}}{\sqrt{5}}.

$$\frac{4}{3}\sqrt{6}-2\sqrt{5}+\frac{\sqrt{9}}{\sqrt{5}}$$

Calculate the square root of 9 and get 3.

$$\frac{4}{3}\sqrt{6}-2\sqrt{5}+\frac{3}{\sqrt{5}}$$

Rationalize the denominator of \frac{3}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{4}{3}\sqrt{6}-2\sqrt{5}+\frac{3\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{4}{3}\sqrt{6}-2\sqrt{5}+\frac{3\sqrt{5}}{5}$$

Combine -2\sqrt{5} and \frac{3\sqrt{5}}{5} to get -\frac{7}{5}\sqrt{5}.

$$\frac{4}{3}\sqrt{6}-\frac{7}{5}\sqrt{5}$$