Solve [(frac{3/4)^{2}:(1/2+5/8)+1]^2-[(1-1/3)^2:(frac{2}{3})^2]^3}{frac{4}{7}:(frac{1}{7}+frac{1}{5})+(frac{7}{3}+frac{1}{5}):19}

$$\frac { [ ( \frac { 3 } { 4 } ) ^ { 2 } : ( \frac { 1 } { 2 } + \frac { 5 } { 8 } ) + 1 ] ^ { 2 } - [ ( 1 - \frac { 1 } { 3 } ) ^ { 2 } : ( \frac { 2 } { 3 } ) ^ { 2 } ] ^ { 3 } } { \frac { 4 } { 7 } : ( \frac { 1 } { 7 } + \frac { 1 } { 5 } ) + ( \frac { 7 } { 3 } + \frac { 1 } { 5 } ) : 19 }$$

Evaluate

$$\frac{25}{36}\approx 0.694444444$$

Solution Steps

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{\left(\frac{\frac{9}{16}}{\frac{1}{2}+\frac{5}{8}}+1\right)^{2}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Add \frac{1}{2} and \frac{5}{8} to get \frac{9}{8}.

$$\frac{\left(\frac{\frac{9}{16}}{\frac{9}{8}}+1\right)^{2}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Divide \frac{9}{16} by \frac{9}{8} by multiplying \frac{9}{16} by the reciprocal of \frac{9}{8}.

$$\frac{\left(\frac{9}{16}\times \frac{8}{9}+1\right)^{2}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Multiply \frac{9}{16} and \frac{8}{9} to get \frac{1}{2}.

$$\frac{\left(\frac{1}{2}+1\right)^{2}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

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

$$\frac{\left(\frac{3}{2}\right)^{2}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\frac{9}{4}-\left(\frac{\left(1-\frac{1}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

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

$$\frac{\frac{9}{4}-\left(\frac{\left(\frac{2}{3}\right)^{2}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{9}{4}-\left(\frac{\frac{4}{9}}{\left(\frac{2}{3}\right)^{2}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{9}{4}-\left(\frac{\frac{4}{9}}{\frac{4}{9}}\right)^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Divide \frac{4}{9} by \frac{4}{9} to get 1.

$$\frac{\frac{9}{4}-1^{3}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Calculate 1 to the power of 3 and get 1.

$$\frac{\frac{9}{4}-1}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Subtract 1 from \frac{9}{4} to get \frac{5}{4}.

$$\frac{\frac{5}{4}}{\frac{\frac{4}{7}}{\frac{1}{7}+\frac{1}{5}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Add \frac{1}{7} and \frac{1}{5} to get \frac{12}{35}.

$$\frac{\frac{5}{4}}{\frac{\frac{4}{7}}{\frac{12}{35}}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Divide \frac{4}{7} by \frac{12}{35} by multiplying \frac{4}{7} by the reciprocal of \frac{12}{35}.

$$\frac{\frac{5}{4}}{\frac{4}{7}\times \frac{35}{12}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Multiply \frac{4}{7} and \frac{35}{12} to get \frac{5}{3}.

$$\frac{\frac{5}{4}}{\frac{5}{3}+\frac{\frac{7}{3}+\frac{1}{5}}{19}}$$

Add \frac{7}{3} and \frac{1}{5} to get \frac{38}{15}.

$$\frac{\frac{5}{4}}{\frac{5}{3}+\frac{\frac{38}{15}}{19}}$$

Express \frac{\frac{38}{15}}{19} as a single fraction.

$$\frac{\frac{5}{4}}{\frac{5}{3}+\frac{38}{15\times 19}}$$

Multiply 15 and 19 to get 285.

$$\frac{\frac{5}{4}}{\frac{5}{3}+\frac{38}{285}}$$

Reduce the fraction \frac{38}{285} to lowest terms by extracting and canceling out 19.

$$\frac{\frac{5}{4}}{\frac{5}{3}+\frac{2}{15}}$$

Add \frac{5}{3} and \frac{2}{15} to get \frac{9}{5}.

$$\frac{\frac{5}{4}}{\frac{9}{5}}$$

Divide \frac{5}{4} by \frac{9}{5} by multiplying \frac{5}{4} by the reciprocal of \frac{9}{5}.

$$\frac{5}{4}\times \frac{5}{9}$$

Multiply \frac{5}{4} and \frac{5}{9} to get \frac{25}{36}.

$$\frac{25}{36}$$

Factor

$$\frac{5 ^ {2}}{2 ^ {2} \cdot 3 ^ {2}} = 0.6944444444444444$$