Solve 2sqrt{10}+3sqrt{11/3}-sqrt{51/5}-2/3

$$2 \sqrt { 10 } + 3 \sqrt { 1 \frac { 1 } { 3 } } - \sqrt { 5 \frac { 1 } { 5 } } - \frac { 2 } { 3 }$$

Evaluate

$$-\frac{\sqrt{130}}{5}+2\sqrt{3}+2\sqrt{10}-\frac{2}{3}\approx 6.841639419$$

Short Solution Steps

Multiply 1 and 3 to get 3.

$$2\sqrt{10}+3\sqrt{\frac{3+1}{3}}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

Add 3 and 1 to get 4.

$$2\sqrt{10}+3\sqrt{\frac{4}{3}}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$2\sqrt{10}+3\times \frac{\sqrt{4}}{\sqrt{3}}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

Calculate the square root of 4 and get 2.

$$2\sqrt{10}+3\times \frac{2}{\sqrt{3}}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

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

$$2\sqrt{10}+3\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

The square of \sqrt{3} is 3.

$$2\sqrt{10}+3\times \frac{2\sqrt{3}}{3}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

Cancel out 3 and 3.

$$2\sqrt{10}+2\sqrt{3}-\sqrt{\frac{5\times 5+1}{5}}-\frac{2}{3}$$

Multiply 5 and 5 to get 25.

$$2\sqrt{10}+2\sqrt{3}-\sqrt{\frac{25+1}{5}}-\frac{2}{3}$$

Add 25 and 1 to get 26.

$$2\sqrt{10}+2\sqrt{3}-\sqrt{\frac{26}{5}}-\frac{2}{3}$$

Rewrite the square root of the division \sqrt{\frac{26}{5}} as the division of square roots \frac{\sqrt{26}}{\sqrt{5}}.

$$2\sqrt{10}+2\sqrt{3}-\frac{\sqrt{26}}{\sqrt{5}}-\frac{2}{3}$$

Rationalize the denominator of \frac{\sqrt{26}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$2\sqrt{10}+2\sqrt{3}-\frac{\sqrt{26}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{2}{3}$$

The square of \sqrt{5} is 5.

$$2\sqrt{10}+2\sqrt{3}-\frac{\sqrt{26}\sqrt{5}}{5}-\frac{2}{3}$$

To multiply \sqrt{26} and \sqrt{5}, multiply the numbers under the square root.

$$2\sqrt{10}+2\sqrt{3}-\frac{\sqrt{130}}{5}-\frac{2}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{\sqrt{130}}{5} times \frac{3}{3}. Multiply \frac{2}{3} times \frac{5}{5}.

$$2\sqrt{10}+2\sqrt{3}-\frac{3\sqrt{130}}{15}-\frac{2\times 5}{15}$$

Since -\frac{3\sqrt{130}}{15} and \frac{2\times 5}{15} have the same denominator, subtract them by subtracting their numerators.

$$2\sqrt{10}+2\sqrt{3}+\frac{-3\sqrt{130}-2\times 5}{15}$$

Do the multiplications in -3\sqrt{130}-2\times 5.

$$2\sqrt{10}+2\sqrt{3}+\frac{-3\sqrt{130}-10}{15}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{10}+2\sqrt{3} times \frac{15}{15}.

$$\frac{15\left(2\sqrt{10}+2\sqrt{3}\right)}{15}+\frac{-3\sqrt{130}-10}{15}$$

Since \frac{15\left(2\sqrt{10}+2\sqrt{3}\right)}{15} and \frac{-3\sqrt{130}-10}{15} have the same denominator, add them by adding their numerators.

$$\frac{15\left(2\sqrt{10}+2\sqrt{3}\right)-3\sqrt{130}-10}{15}$$

Do the multiplications in 15\left(2\sqrt{10}+2\sqrt{3}\right)-3\sqrt{130}-10.

$$\frac{30\sqrt{10}+30\sqrt{3}-3\sqrt{130}-10}{15}$$

Factor

$$\frac{30 \sqrt{3} + 30 \sqrt{10} - 3 \sqrt{130} - 10}{15} = 6.841639418609572$$