Solve sqrt{30}+3sqrt{13/5}*2/3sqrt{62/3}

$$\sqrt { 30 } + 3 \sqrt { 1 \frac { 3 } { 5 } } \times \frac { 2 } { 3 } \sqrt { 6 \frac { 2 } { 3 } }$$

Evaluate

$$\frac{8\sqrt{6}}{3}+\sqrt{30}\approx 12.009198222$$

Short Solution Steps

Multiply 1 and 5 to get 5.

$$\sqrt{30}+3\sqrt{\frac{5+3}{5}}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

Add 5 and 3 to get 8.

$$\sqrt{30}+3\sqrt{\frac{8}{5}}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$\sqrt{30}+3\times \frac{\sqrt{8}}{\sqrt{5}}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\sqrt{30}+3\times \frac{2\sqrt{2}}{\sqrt{5}}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

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

$$\sqrt{30}+3\times \frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

The square of \sqrt{5} is 5.

$$\sqrt{30}+3\times \frac{2\sqrt{2}\sqrt{5}}{5}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

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

$$\sqrt{30}+3\times \frac{2\sqrt{10}}{5}\times \frac{2}{3}\sqrt{\frac{6\times 3+2}{3}}$$

Cancel out 3 and 3.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\sqrt{\frac{6\times 3+2}{3}}$$

Multiply 6 and 3 to get 18.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\sqrt{\frac{18+2}{3}}$$

Add 18 and 2 to get 20.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\sqrt{\frac{20}{3}}$$

Rewrite the square root of the division \sqrt{\frac{20}{3}} as the division of square roots \frac{\sqrt{20}}{\sqrt{3}}.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\times \frac{\sqrt{20}}{\sqrt{3}}$$

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}.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\times \frac{2\sqrt{5}}{\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\times \frac{2\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\times \frac{2\sqrt{5}\sqrt{3}}{3}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\sqrt{30}+2\times \frac{2\sqrt{10}}{5}\times \frac{2\sqrt{15}}{3}$$

Express 2\times \frac{2\sqrt{10}}{5} as a single fraction.

$$\sqrt{30}+\frac{2\times 2\sqrt{10}}{5}\times \frac{2\sqrt{15}}{3}$$

Multiply \frac{2\times 2\sqrt{10}}{5} times \frac{2\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{30}+\frac{2\times 2\sqrt{10}\times 2\sqrt{15}}{5\times 3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{30} times \frac{5\times 3}{5\times 3}.

$$\frac{\sqrt{30}\times 5\times 3}{5\times 3}+\frac{2\times 2\sqrt{10}\times 2\sqrt{15}}{5\times 3}$$

Since \frac{\sqrt{30}\times 5\times 3}{5\times 3} and \frac{2\times 2\sqrt{10}\times 2\sqrt{15}}{5\times 3} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{30}\times 5\times 3+2\times 2\sqrt{10}\times 2\sqrt{15}}{5\times 3}$$

Do the multiplications in \sqrt{30}\times 5\times 3+2\times 2\sqrt{10}\times 2\sqrt{15}.

$$\frac{15\sqrt{30}+40\sqrt{6}}{5\times 3}$$

Expand 5\times 3.

$$\frac{15\sqrt{30}+40\sqrt{6}}{15}$$