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

$$\left. \begin{array} { l } { \sqrt { 5 } - \sqrt { 6 } } \\ { - \sqrt { 20 } + \sqrt { \frac { 2 } { 3 } } } \\ { + \sqrt { \frac { 9 } { 5 } } } \end{array} \right.$$

Evaluate

$$-\frac{2\sqrt{5}}{5}-\frac{2\sqrt{6}}{3}\approx -2.527420353$$

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}.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Combine -\sqrt{6} and \frac{\sqrt{6}}{3} to get -\frac{2}{3}\sqrt{6}.

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

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

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

Calculate the square root of 9 and get 3.

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

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

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

The square of \sqrt{5} is 5.

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

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

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