Solve sqrt{11/3}+sqrt{11/2}

$$\sqrt { 1 \frac { 1 } { 3 } } + \sqrt { 1 \frac { 1 } { 2 } }$$

Evaluate

$$\frac{\sqrt{3}\left(3\sqrt{2}+4\right)}{6}\approx 2.37944541$$

Short Solution Steps

Multiply 1 and 3 to get 3.

$$\sqrt{\frac{3+1}{3}}+\sqrt{\frac{1\times 2+1}{2}}$$

Add 3 and 1 to get 4.

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

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

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

Calculate the square root of 4 and get 2.

$$\frac{2}{\sqrt{3}}+\sqrt{\frac{1\times 2+1}{2}}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}+\sqrt{\frac{1\times 2+1}{2}}$$

Multiply 1 and 2 to get 2.

$$\frac{2\sqrt{3}}{3}+\sqrt{\frac{2+1}{2}}$$

Add 2 and 1 to get 3.

$$\frac{2\sqrt{3}}{3}+\sqrt{\frac{3}{2}}$$

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

$$\frac{2\sqrt{3}}{3}+\frac{\sqrt{3}}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{3}}{3}+\frac{\sqrt{3}\sqrt{2}}{2}$$

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

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

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

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

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

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

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

$$\frac{4\sqrt{3}+3\sqrt{6}}{6}$$