Solve {(1/6)^{2}+[(1/3)^{2}-(1/3-1/6)^{2}]}*{[frac{1}{2}+(10^0-frac{1}{3}-frac{4}{6})^4]-frac{2}{3^2}-(frac{1}{2}-frac{1}{3})^2}

$$\{ ( \frac { 1 } { 6 } ) ^ { 2 } + [ ( \frac { 1 } { 3 } ) ^ { 2 } - ( \frac { 1 } { 3 } - \frac { 1 } { 6 } ) ^ { 2 } ] \} \times \{ [ \frac { 1 } { 2 } + ( 10 ^ { 0 } - \frac { 1 } { 3 } - \frac { 4 } { 6 } ) ^ { 4 } ] - \frac { 2 } { 3 ^ { 2 } } - ( \frac { 1 } { 2 } - \frac { 1 } { 3 } ) ^ { 2 } \}$$

Evaluate

$$\frac{1}{36}\approx 0.027777778$$

Solution Steps

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

$$\left(\frac{1}{36}+\left(\frac{1}{3}\right)^{2}-\left(\frac{1}{3}-\frac{1}{6}\right)^{2}\right)\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\left(\frac{1}{36}+\frac{1}{9}-\left(\frac{1}{3}-\frac{1}{6}\right)^{2}\right)\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

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

$$\left(\frac{1}{36}+\frac{1}{9}-\left(\frac{1}{6}\right)^{2}\right)\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

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

$$\left(\frac{1}{36}+\frac{1}{9}-\frac{1}{36}\right)\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Subtract \frac{1}{36} from \frac{1}{9} to get \frac{1}{12}.

$$\left(\frac{1}{36}+\frac{1}{12}\right)\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

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

$$\frac{1}{9}\left(\frac{1}{2}+\left(10^{0}-\frac{1}{3}-\frac{4}{6}\right)^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Calculate 10 to the power of 0 and get 1.

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

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

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

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

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

Subtract \frac{2}{3} from \frac{2}{3} to get 0.

$$\frac{1}{9}\left(\frac{1}{2}+0^{4}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Calculate 0 to the power of 4 and get 0.

$$\frac{1}{9}\left(\frac{1}{2}+0-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Add \frac{1}{2} and 0 to get \frac{1}{2}.

$$\frac{1}{9}\left(\frac{1}{2}-\frac{2}{3^{2}}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{9}\left(\frac{1}{2}-\frac{2}{9}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

Subtract \frac{2}{9} from \frac{1}{2} to get \frac{5}{18}.

$$\frac{1}{9}\left(\frac{5}{18}-\left(\frac{1}{2}-\frac{1}{3}\right)^{2}\right)$$

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

$$\frac{1}{9}\left(\frac{5}{18}-\left(\frac{1}{6}\right)^{2}\right)$$

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

$$\frac{1}{9}\left(\frac{5}{18}-\frac{1}{36}\right)$$

Subtract \frac{1}{36} from \frac{5}{18} to get \frac{1}{4}.

$$\frac{1}{9}\times \frac{1}{4}$$

Multiply \frac{1}{9} and \frac{1}{4} to get \frac{1}{36}.

$$\frac{1}{36}$$

Factor

$$\frac{1}{2 ^ {2} \cdot 3 ^ {2}} = 0.027777777777777776$$