Solve sqrt{11/3}*frac{(sqrt{3}+sqrt{2})^2}{sqrt{22}}+frac{sqrt{6}}{3sqrt{(-2)^2}}

$$\sqrt { \frac { 11 } { 3 } } \times \frac { ( \sqrt { 3 } + \sqrt { 2 } ) ^ { 2 } } { \sqrt { 22 } } + \frac { \sqrt { 6 } } { 3 \sqrt { ( - 2 ) ^ { 2 } } }$$

Evaluate

$$\sqrt{6}+2\approx 4.449489743$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{11}{3}} as the division of square roots \frac{\sqrt{11}}{\sqrt{3}}.

$$\frac{\sqrt{11}}{\sqrt{3}}\times \frac{\left(\sqrt{3}+\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{11}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times \frac{\left(\sqrt{3}+\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{11}\sqrt{3}}{3}\times \frac{\left(\sqrt{3}+\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

To multiply \sqrt{11} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{33}}{3}\times \frac{\left(\sqrt{3}+\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+\sqrt{2}\right)^{2}.

$$\frac{\sqrt{33}}{3}\times \frac{\left(\sqrt{3}\right)^{2}+2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{33}}{3}\times \frac{3+2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

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

$$\frac{\sqrt{33}}{3}\times \frac{3+2\sqrt{6}+\left(\sqrt{2}\right)^{2}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{33}}{3}\times \frac{3+2\sqrt{6}+2}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Add 3 and 2 to get 5.

$$\frac{\sqrt{33}}{3}\times \frac{5+2\sqrt{6}}{\sqrt{22}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Rationalize the denominator of \frac{5+2\sqrt{6}}{\sqrt{22}} by multiplying numerator and denominator by \sqrt{22}.

$$\frac{\sqrt{33}}{3}\times \frac{\left(5+2\sqrt{6}\right)\sqrt{22}}{\left(\sqrt{22}\right)^{2}}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

The square of \sqrt{22} is 22.

$$\frac{\sqrt{33}}{3}\times \frac{\left(5+2\sqrt{6}\right)\sqrt{22}}{22}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Multiply \frac{\sqrt{33}}{3} times \frac{\left(5+2\sqrt{6}\right)\sqrt{22}}{22} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{3\times 22}+\frac{\sqrt{6}}{3\sqrt{\left(-2\right)^{2}}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{3\times 22}+\frac{\sqrt{6}}{3\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{3\times 22}+\frac{\sqrt{6}}{3\times 2}$$

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{3\times 22}+\frac{\sqrt{6}}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\times 22 and 6 is 66. Multiply \frac{\sqrt{6}}{6} times \frac{11}{11}.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{66}+\frac{11\sqrt{6}}{66}$$

Since \frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}}{66} and \frac{11\sqrt{6}}{66} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}+11\sqrt{6}}{66}$$

Do the multiplications in \sqrt{33}\left(5+2\sqrt{6}\right)\sqrt{22}+11\sqrt{6}.

$$\frac{55\sqrt{6}+132+11\sqrt{6}}{66}$$

Do the calculations in 55\sqrt{6}+132+11\sqrt{6}.

$$\frac{66\sqrt{6}+132}{66}$$

Divide each term of 66\sqrt{6}+132 by 66 to get \sqrt{6}+2.

$$\sqrt{6}+2$$

Factor

$$\sqrt{6} + 2 = 4.449489743$$