Solve 2/3-(1+1/2)^{2}(1-1/2)^{2}*(-2/3)^{2}+[(-frac{1}{6})^2*(frac{1}{6})^3]^2:(frac{1}{6})^8-frac{1}{6}

$$\frac { 2 } { 3 } - ( 1 + \frac { 1 } { 2 } ) ^ { 2 } ( 1 - \frac { 1 } { 2 } ) ^ { 2 } \cdot ( - \frac { 2 } { 3 } ) ^ { 2 } + [ ( - \frac { 1 } { 6 } ) ^ { 2 } \cdot ( \frac { 1 } { 6 } ) ^ { 3 } ] ^ { 2 } : ( \frac { 1 } { 6 } ) ^ { 8 } - \frac { 1 } { 6 }$$

Evaluate

$$\frac{5}{18}\approx 0.277777778$$

Solution Steps

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

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

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

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

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

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

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

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

Multiply \frac{9}{4} and \frac{1}{4} to get \frac{9}{16}.

$$\frac{2}{3}-\frac{9}{16}\left(-\frac{2}{3}\right)^{2}+\frac{\left(\left(-\frac{1}{6}\right)^{2}\times \left(\frac{1}{6}\right)^{3}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

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

$$\frac{2}{3}-\frac{9}{16}\times \frac{4}{9}+\frac{\left(\left(-\frac{1}{6}\right)^{2}\times \left(\frac{1}{6}\right)^{3}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Multiply \frac{9}{16} and \frac{4}{9} to get \frac{1}{4}.

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

Subtract \frac{1}{4} from \frac{2}{3} to get \frac{5}{12}.

$$\frac{5}{12}+\frac{\left(\left(-\frac{1}{6}\right)^{2}\times \left(\frac{1}{6}\right)^{3}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Calculate -\frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$\frac{5}{12}+\frac{\left(\frac{1}{36}\times \left(\frac{1}{6}\right)^{3}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Calculate \frac{1}{6} to the power of 3 and get \frac{1}{216}.

$$\frac{5}{12}+\frac{\left(\frac{1}{36}\times \frac{1}{216}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Multiply \frac{1}{36} and \frac{1}{216} to get \frac{1}{7776}.

$$\frac{5}{12}+\frac{\left(\frac{1}{7776}\right)^{2}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Calculate \frac{1}{7776} to the power of 2 and get \frac{1}{60466176}.

$$\frac{5}{12}+\frac{\frac{1}{60466176}}{\left(\frac{1}{6}\right)^{8}}-\frac{1}{6}$$

Calculate \frac{1}{6} to the power of 8 and get \frac{1}{1679616}.

$$\frac{5}{12}+\frac{\frac{1}{60466176}}{\frac{1}{1679616}}-\frac{1}{6}$$

Divide \frac{1}{60466176} by \frac{1}{1679616} by multiplying \frac{1}{60466176} by the reciprocal of \frac{1}{1679616}.

$$\frac{5}{12}+\frac{1}{60466176}\times 1679616-\frac{1}{6}$$

Multiply \frac{1}{60466176} and 1679616 to get \frac{1}{36}.

$$\frac{5}{12}+\frac{1}{36}-\frac{1}{6}$$

Add \frac{5}{12} and \frac{1}{36} to get \frac{4}{9}.

$$\frac{4}{9}-\frac{1}{6}$$

Subtract \frac{1}{6} from \frac{4}{9} to get \frac{5}{18}.

$$\frac{5}{18}$$

Factor

$$\frac{5}{2 \cdot 3 ^ {2}} = 0.2777777777777778$$