Solve sqrt{[7/5*1/10:14/5*(7/12+frac{2}{3})^2+(frac{3}{4})^2:(frac{3}{4})^2]:{[1:frac{1}{2}-(frac{16}{5}-frac{7}{10}-frac{3}{4})]*(frac{3}{2})^2}:frac{1}{2}}

$$\sqrt { [ \frac { 7 } { 5 } \times \frac { 1 } { 10 } : \frac { 14 } { 5 } \times ( \frac { 7 } { 12 } + \frac { 2 } { 3 } ) ^ { 2 } + ( \frac { 3 } { 4 } ) ^ { 2 } : ( \frac { 3 } { 4 } ) ^ { 2 } ] : \{ [ 1 : \frac { 1 } { 2 } - ( \frac { 16 } { 5 } - \frac { 7 } { 10 } - \frac { 3 } { 4 } ) ] \times ( \frac { 3 } { 2 } ) ^ { 2 } \} : \frac { 1 } { 2 } }$$

Evaluate

$$\frac{\sqrt{138}}{6}\approx 1.957890021$$

Solution Steps

Divide \left(\frac{3}{4}\right)^{2} by \left(\frac{3}{4}\right)^{2} to get 1.

$$\sqrt{\frac{\frac{\frac{\frac{7}{5}\times \frac{1}{10}}{\frac{14}{5}}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}{\frac{1}{2}}}$$

Divide \frac{\frac{\frac{7}{5}\times \frac{1}{10}}{\frac{14}{5}}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}} by \frac{1}{2} by multiplying \frac{\frac{\frac{7}{5}\times \frac{1}{10}}{\frac{14}{5}}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}} by the reciprocal of \frac{1}{2}.

$$\sqrt{\frac{\left(\frac{\frac{7}{5}\times \frac{1}{10}}{\frac{14}{5}}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Multiply \frac{7}{5} and \frac{1}{10} to get \frac{7}{50}.

$$\sqrt{\frac{\left(\frac{\frac{7}{50}}{\frac{14}{5}}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Divide \frac{7}{50} by \frac{14}{5} by multiplying \frac{7}{50} by the reciprocal of \frac{14}{5}.

$$\sqrt{\frac{\left(\frac{7}{50}\times \frac{5}{14}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Multiply \frac{7}{50} and \frac{5}{14} to get \frac{1}{20}.

$$\sqrt{\frac{\left(\frac{1}{20}\left(\frac{7}{12}+\frac{2}{3}\right)^{2}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Add \frac{7}{12} and \frac{2}{3} to get \frac{5}{4}.

$$\sqrt{\frac{\left(\frac{1}{20}\times \left(\frac{5}{4}\right)^{2}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Calculate \frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$\sqrt{\frac{\left(\frac{1}{20}\times \frac{25}{16}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Multiply \frac{1}{20} and \frac{25}{16} to get \frac{5}{64}.

$$\sqrt{\frac{\left(\frac{5}{64}+1\right)\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Add \frac{5}{64} and 1 to get \frac{69}{64}.

$$\sqrt{\frac{\frac{69}{64}\times 2}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Multiply \frac{69}{64} and 2 to get \frac{69}{32}.

$$\sqrt{\frac{\frac{69}{32}}{\left(\frac{1}{\frac{1}{2}}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

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

$$\sqrt{\frac{\frac{69}{32}}{\left(1\times 2-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Multiply 1 and 2 to get 2.

$$\sqrt{\frac{\frac{69}{32}}{\left(2-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Subtract \frac{7}{10} from \frac{16}{5} to get \frac{5}{2}.

$$\sqrt{\frac{\frac{69}{32}}{\left(2-\left(\frac{5}{2}-\frac{3}{4}\right)\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Subtract \frac{3}{4} from \frac{5}{2} to get \frac{7}{4}.

$$\sqrt{\frac{\frac{69}{32}}{\left(2-\frac{7}{4}\right)\times \left(\frac{3}{2}\right)^{2}}}$$

Subtract \frac{7}{4} from 2 to get \frac{1}{4}.

$$\sqrt{\frac{\frac{69}{32}}{\frac{1}{4}\times \left(\frac{3}{2}\right)^{2}}}$$

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

$$\sqrt{\frac{\frac{69}{32}}{\frac{1}{4}\times \frac{9}{4}}}$$

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

$$\sqrt{\frac{\frac{69}{32}}{\frac{9}{16}}}$$

Divide \frac{69}{32} by \frac{9}{16} by multiplying \frac{69}{32} by the reciprocal of \frac{9}{16}.

$$\sqrt{\frac{69}{32}\times \frac{16}{9}}$$

Multiply \frac{69}{32} and \frac{16}{9} to get \frac{23}{6}.

$$\sqrt{\frac{23}{6}}$$

Rewrite the square root of the division \sqrt{\frac{23}{6}} as the division of square roots \frac{\sqrt{23}}{\sqrt{6}}.

$$\frac{\sqrt{23}}{\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{23}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{23}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{23}\sqrt{6}}{6}$$

To multiply \sqrt{23} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{138}}{6}$$