Solve 6sqrt{13/5}+4sqrt{22/5}div2sqrt{11/3}

$$6 \sqrt { 1 \frac { 3 } { 5 } } + 4 \sqrt { 2 \frac { 2 } { 5 } } \div 2 \sqrt { 1 \frac { 1 } { 3 } }$$

Evaluate

$$\frac{8\sqrt{5}+12\sqrt{10}}{5}\approx 11.167175148$$

Short Solution Steps

Multiply 1 and 5 to get 5.

$$6\sqrt{\frac{5+3}{5}}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Add 5 and 3 to get 8.

$$6\sqrt{\frac{8}{5}}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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

$$6\times \frac{\sqrt{8}}{\sqrt{5}}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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}.

$$6\times \frac{2\sqrt{2}}{\sqrt{5}}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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

$$6\times \frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

The square of \sqrt{5} is 5.

$$6\times \frac{2\sqrt{2}\sqrt{5}}{5}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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

$$6\times \frac{2\sqrt{10}}{5}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Express 6\times \frac{2\sqrt{10}}{5} as a single fraction.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{\frac{2\times 5+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Multiply 2 and 5 to get 10.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{\frac{10+2}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Add 10 and 2 to get 12.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{\frac{12}{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{12}{5}} as the division of square roots \frac{\sqrt{12}}{\sqrt{5}}.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times \frac{\sqrt{12}}{\sqrt{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times \frac{2\sqrt{3}}{\sqrt{5}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times \frac{2\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

The square of \sqrt{5} is 5.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times \frac{2\sqrt{3}\sqrt{5}}{5}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

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

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times \frac{2\sqrt{15}}{5}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Express 4\times \frac{2\sqrt{15}}{5} as a single fraction.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{\frac{4\times 2\sqrt{15}}{5}}{2}\sqrt{\frac{1\times 3+1}{3}}$$

Express \frac{\frac{4\times 2\sqrt{15}}{5}}{2} as a single fraction.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\times 2\sqrt{15}}{5\times 2}\sqrt{\frac{1\times 3+1}{3}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{2\times 2\sqrt{15}}{5}\sqrt{\frac{1\times 3+1}{3}}$$

Multiply 2 and 2 to get 4.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\sqrt{\frac{1\times 3+1}{3}}$$

Multiply 1 and 3 to get 3.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\sqrt{\frac{3+1}{3}}$$

Add 3 and 1 to get 4.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\sqrt{\frac{4}{3}}$$

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

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\times \frac{\sqrt{4}}{\sqrt{3}}$$

Calculate the square root of 4 and get 2.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\times \frac{2}{\sqrt{3}}$$

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

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}}{5}\times \frac{2\sqrt{3}}{3}$$

Multiply \frac{4\sqrt{15}}{5} times \frac{2\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{6\times 2\sqrt{10}}{5}+\frac{4\sqrt{15}\times 2\sqrt{3}}{5\times 3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 5\times 3 is 3\times 5. Multiply \frac{6\times 2\sqrt{10}}{5} times \frac{3}{3}.

$$\frac{3\times 6\times 2\sqrt{10}}{3\times 5}+\frac{4\sqrt{15}\times 2\sqrt{3}}{3\times 5}$$

Since \frac{3\times 6\times 2\sqrt{10}}{3\times 5} and \frac{4\sqrt{15}\times 2\sqrt{3}}{3\times 5} have the same denominator, add them by adding their numerators.

$$\frac{3\times 6\times 2\sqrt{10}+4\sqrt{15}\times 2\sqrt{3}}{3\times 5}$$

Do the multiplications in 3\times 6\times 2\sqrt{10}+4\sqrt{15}\times 2\sqrt{3}.

$$\frac{36\sqrt{10}+24\sqrt{5}}{3\times 5}$$

Expand 3\times 5.

$$\frac{36\sqrt{10}+24\sqrt{5}}{15}$$

Factor

$$\frac{4 {(2 \sqrt{5} + 3 \sqrt{10})}}{5} = 11.167175148403775$$