Solve 208sqrt{{left(1/2right)}^2+{left(frac{4/3+frac{1/6right)}^5}{{left(1+1/2right)}^3}div1-frac{1}{6}-frac{2}{3}}{frac{frac{2}{3}-frac{3}{7}}{frac{15}{7}}+frac{4}{9}}}

$$208 \sqrt{ { \left( \frac{ 1 }{ 2 } \right) }^{ 2 } + \frac{ \frac{ { \left( \frac{ 4 }{ 3 } + \frac{ 1 }{ 6 } \right) }^{ 5 } }{ { \left(1+ \frac{ 1 }{ 2 } \right) }^{ 3 } } \div 1- \frac{ 1 }{ 6 } - \frac{ 2 }{ 3 } }{ \frac{ \frac{ 2 }{ 3 } - \frac{ 3 }{ 7 } }{ \frac{ 15 }{ 7 } } + \frac{ 4 }{ 9 } } }$$

Evaluate

$$\frac{208\sqrt{70}}{5}\approx 348.050571038$$

Solution Steps

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{\left(\frac{4}{3}+\frac{1}{6}\right)^{5}}{\left(1+\frac{1}{2}\right)^{3}}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Add \frac{4}{3} and \frac{1}{6} to get \frac{3}{2}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{\left(\frac{3}{2}\right)^{5}}{\left(1+\frac{1}{2}\right)^{3}}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Calculate \frac{3}{2} to the power of 5 and get \frac{243}{32}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{\frac{243}{32}}{\left(1+\frac{1}{2}\right)^{3}}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{\frac{243}{32}}{\left(\frac{3}{2}\right)^{3}}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Calculate \frac{3}{2} to the power of 3 and get \frac{27}{8}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{\frac{243}{32}}{\frac{27}{8}}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Divide \frac{243}{32} by \frac{27}{8} by multiplying \frac{243}{32} by the reciprocal of \frac{27}{8}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{243}{32}\times \frac{8}{27}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Multiply \frac{243}{32} and \frac{8}{27} to get \frac{9}{4}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{\frac{9}{4}}{1}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Anything divided by one gives itself.

$$208\sqrt{\frac{1}{4}+\frac{\frac{9}{4}-\frac{1}{6}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Subtract \frac{1}{6} from \frac{9}{4} to get \frac{25}{12}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{25}{12}-\frac{2}{3}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Subtract \frac{2}{3} from \frac{25}{12} to get \frac{17}{12}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{17}{12}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Subtract \frac{3}{7} from \frac{2}{3} to get \frac{5}{21}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{17}{12}}{\frac{\frac{5}{21}}{\frac{15}{7}}+\frac{4}{9}}}$$

Divide \frac{5}{21} by \frac{15}{7} by multiplying \frac{5}{21} by the reciprocal of \frac{15}{7}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{17}{12}}{\frac{5}{21}\times \frac{7}{15}+\frac{4}{9}}}$$

Multiply \frac{5}{21} and \frac{7}{15} to get \frac{1}{9}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{17}{12}}{\frac{1}{9}+\frac{4}{9}}}$$

Add \frac{1}{9} and \frac{4}{9} to get \frac{5}{9}.

$$208\sqrt{\frac{1}{4}+\frac{\frac{17}{12}}{\frac{5}{9}}}$$

Divide \frac{17}{12} by \frac{5}{9} by multiplying \frac{17}{12} by the reciprocal of \frac{5}{9}.

$$208\sqrt{\frac{1}{4}+\frac{17}{12}\times \frac{9}{5}}$$

Multiply \frac{17}{12} and \frac{9}{5} to get \frac{51}{20}.

$$208\sqrt{\frac{1}{4}+\frac{51}{20}}$$

Add \frac{1}{4} and \frac{51}{20} to get \frac{14}{5}.

$$208\sqrt{\frac{14}{5}}$$

Rewrite the square root of the division \sqrt{\frac{14}{5}} as the division of square roots \frac{\sqrt{14}}{\sqrt{5}}.

$$208\times \frac{\sqrt{14}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$208\times \frac{\sqrt{14}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$208\times \frac{\sqrt{14}\sqrt{5}}{5}$$

To multiply \sqrt{14} and \sqrt{5}, multiply the numbers under the square root.

$$208\times \frac{\sqrt{70}}{5}$$

Express 208\times \frac{\sqrt{70}}{5} as a single fraction.

$$\frac{208\sqrt{70}}{5}$$