Solve sqrt{(1/2)^2+[(4/3+1/6)^3:(1+1/2)^3-(1-frac{1}{6}-frac{2}{3})]:[(frac{2}{3}-frac{3}{7}):frac{15}{7}+frac{4}{9}}]

$$\sqrt { ( \frac { 1 } { 2 } ) ^ { 2 } + [ ( \frac { 4 } { 3 } + \frac { 1 } { 6 } ) ^ { 3 } : ( 1 + \frac { 1 } { 2 } ) ^ { 3 } - ( 1 - \frac { 1 } { 6 } - \frac { 2 } { 3 } ) ] : [ ( \frac { 2 } { 3 } - \frac { 3 } { 7 } ) : \frac { 15 } { 7 } + \frac { 4 } { 9 } } ]$$

Evaluate

$$\frac{\sqrt{7}}{2}\approx 1.322875656$$

Solution Steps

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

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

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

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

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

$$\sqrt{\frac{1}{4}+\frac{\frac{\frac{27}{8}}{\left(1+\frac{1}{2}\right)^{3}}-\left(1-\frac{1}{6}-\frac{2}{3}\right)}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{\frac{27}{8}}{\left(\frac{3}{2}\right)^{3}}-\left(1-\frac{1}{6}-\frac{2}{3}\right)}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{\frac{27}{8}}{\frac{27}{8}}-\left(1-\frac{1}{6}-\frac{2}{3}\right)}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

Divide \frac{27}{8} by \frac{27}{8} to get 1.

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

Subtract \frac{1}{6} from 1 to get \frac{5}{6}.

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

Subtract \frac{2}{3} from \frac{5}{6} to get \frac{1}{6}.

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

Subtract \frac{1}{6} from 1 to get \frac{5}{6}.

$$\sqrt{\frac{1}{4}+\frac{\frac{5}{6}}{\frac{\frac{2}{3}-\frac{3}{7}}{\frac{15}{7}}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{5}{6}}{\frac{\frac{5}{21}}{\frac{15}{7}}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{5}{6}}{\frac{5}{21}\times \frac{7}{15}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{5}{6}}{\frac{1}{9}+\frac{4}{9}}}$$

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

$$\sqrt{\frac{1}{4}+\frac{\frac{5}{6}}{\frac{5}{9}}}$$

Divide \frac{5}{6} by \frac{5}{9} by multiplying \frac{5}{6} by the reciprocal of \frac{5}{9}.

$$\sqrt{\frac{1}{4}+\frac{5}{6}\times \frac{9}{5}}$$

Multiply \frac{5}{6} and \frac{9}{5} to get \frac{3}{2}.

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

Add \frac{1}{4} and \frac{3}{2} to get \frac{7}{4}.

$$\sqrt{\frac{7}{4}}$$

Rewrite the square root of the division \sqrt{\frac{7}{4}} as the division of square roots \frac{\sqrt{7}}{\sqrt{4}}.

$$\frac{\sqrt{7}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{7}}{2}$$