Solve 1/3(sqrt{frac{8/3}+2}{2/3})^2+4(sqrt{frac{8/3}+2}{frac{2}{3}})

$$\frac { 1 } { 3 } ( \frac { \sqrt { \frac { 8 } { 3 } } + 2 } { \frac { 2 } { 3 } } ) ^ { 2 } + 4 ( \frac { \sqrt { \frac { 8 } { 3 } } + 2 } { \frac { 2 } { 3 } } )$$

Evaluate

$$6\sqrt{6}+17\approx 31.696938457$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{1}{3}\times \left(\frac{\frac{\sqrt{8}}{\sqrt{3}}+2}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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}.

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{2}}{\sqrt{3}}+2}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+2}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

The square of \sqrt{3} is 3.

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{2}\sqrt{3}}{3}+2}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{6}}{3}+2}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{6}}{3}+\frac{2\times 3}{3}}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{6}+2\times 3}{3}}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

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

$$\frac{1}{3}\times \left(\frac{\frac{2\sqrt{6}+6}{3}}{\frac{2}{3}}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Divide \frac{2\sqrt{6}+6}{3} by \frac{2}{3} by multiplying \frac{2\sqrt{6}+6}{3} by the reciprocal of \frac{2}{3}.

$$\frac{1}{3}\times \left(\frac{\left(2\sqrt{6}+6\right)\times 3}{3\times 2}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{1}{3}\times \left(\frac{2\sqrt{6}+6}{2}\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Divide each term of 2\sqrt{6}+6 by 2 to get \sqrt{6}+3.

$$\frac{1}{3}\left(\sqrt{6}+3\right)^{2}+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{6}+3\right)^{2}.

$$\frac{1}{3}\left(\left(\sqrt{6}\right)^{2}+6\sqrt{6}+9\right)+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

The square of \sqrt{6} is 6.

$$\frac{1}{3}\left(6+6\sqrt{6}+9\right)+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Add 6 and 9 to get 15.

$$\frac{1}{3}\left(15+6\sqrt{6}\right)+4\times \frac{\sqrt{\frac{8}{3}}+2}{\frac{2}{3}}$$

Use the distributive property to multiply \frac{1}{3} by 15+6\sqrt{6}.

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

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

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

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}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

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

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

Divide \frac{2\sqrt{6}+6}{3} by \frac{2}{3} by multiplying \frac{2\sqrt{6}+6}{3} by the reciprocal of \frac{2}{3}.

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

Cancel out 3 in both numerator and denominator.

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

Divide each term of 2\sqrt{6}+6 by 2 to get \sqrt{6}+3.

$$5+2\sqrt{6}+4\left(\sqrt{6}+3\right)$$

Use the distributive property to multiply 4 by \sqrt{6}+3.

$$5+2\sqrt{6}+4\sqrt{6}+12$$

Combine 2\sqrt{6} and 4\sqrt{6} to get 6\sqrt{6}.

$$5+6\sqrt{6}+12$$

Add 5 and 12 to get 17.

$$17+6\sqrt{6}$$

Factor

$$6 \sqrt{6} + 17 = 31.696938457$$