Solve sqrt{{[1/2*(5/4-1/2)]^2*[(1/2+frac{1}{3}-frac{1}{6})^2:(frac{1}{6}+frac{1}{2})^2]*(1+frac{1}{3})+(2-frac{3}{16})}:frac{1}{2}}

$$\sqrt { \{ [ \frac { 1 } { 2 } \times ( \frac { 5 } { 4 } - \frac { 1 } { 2 } ) ] ^ { 2 } \times [ ( \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 6 } ) ^ { 2 } : ( \frac { 1 } { 6 } + \frac { 1 } { 2 } ) ^ { 2 } ] \times ( 1 + \frac { 1 } { 3 } ) + ( 2 - \frac { 3 } { 16 } ) \} : \frac { 1 } { 2 } }$$

Evaluate

$$2$$

Solution Steps

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

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

Multiply \frac{1}{2} and \frac{3}{4} to get \frac{3}{8}.

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

Calculate \frac{3}{8} to the power of 2 and get \frac{9}{64}.

$$\sqrt{\frac{\frac{9}{64}\times \frac{\left(\frac{1}{2}+\frac{1}{3}-\frac{1}{6}\right)^{2}}{\left(\frac{1}{6}+\frac{1}{2}\right)^{2}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

Add \frac{1}{2} and \frac{1}{3} to get \frac{5}{6}.

$$\sqrt{\frac{\frac{9}{64}\times \frac{\left(\frac{5}{6}-\frac{1}{6}\right)^{2}}{\left(\frac{1}{6}+\frac{1}{2}\right)^{2}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times \frac{\left(\frac{2}{3}\right)^{2}}{\left(\frac{1}{6}+\frac{1}{2}\right)^{2}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times \frac{\frac{4}{9}}{\left(\frac{1}{6}+\frac{1}{2}\right)^{2}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times \frac{\frac{4}{9}}{\left(\frac{2}{3}\right)^{2}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times \frac{\frac{4}{9}}{\frac{4}{9}}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times 1\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

Multiply \frac{9}{64} and 1 to get \frac{9}{64}.

$$\sqrt{\frac{\frac{9}{64}\left(1+\frac{1}{3}\right)+2-\frac{3}{16}}{\frac{1}{2}}}$$

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

$$\sqrt{\frac{\frac{9}{64}\times \frac{4}{3}+2-\frac{3}{16}}{\frac{1}{2}}}$$

Multiply \frac{9}{64} and \frac{4}{3} to get \frac{3}{16}.

$$\sqrt{\frac{\frac{3}{16}+2-\frac{3}{16}}{\frac{1}{2}}}$$

Add \frac{3}{16} and 2 to get \frac{35}{16}.

$$\sqrt{\frac{\frac{35}{16}-\frac{3}{16}}{\frac{1}{2}}}$$

Subtract \frac{3}{16} from \frac{35}{16} to get 2.

$$\sqrt{\frac{2}{\frac{1}{2}}}$$

Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.

$$\sqrt{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$2$$

Factor

$$2$$