Solve sqrt{[(2/5+1-1/2)*10/9+2]:[10/3*frac{6}{5}*(frac{7}{4}*frac{1}{7}+frac{1}{4})]}

$$\sqrt { [ ( \frac { 2 } { 5 } + 1 - \frac { 1 } { 2 } ) \cdot \frac { 10 } { 9 } + 2 ] : [ \frac { 10 } { 3 } \cdot \frac { 6 } { 5 } \cdot ( \frac { 7 } { 4 } \cdot \frac { 1 } { 7 } + \frac { 1 } { 4 } ) ] }$$

Evaluate

$$\frac{\sqrt{6}}{2}\approx 1.224744871$$

Solution Steps

Convert 1 to fraction \frac{5}{5}.

$$\sqrt{\frac{\left(\frac{2}{5}+\frac{5}{5}-\frac{1}{2}\right)\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Since \frac{2}{5} and \frac{5}{5} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(\frac{2+5}{5}-\frac{1}{2}\right)\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Add 2 and 5 to get 7.

$$\sqrt{\frac{\left(\frac{7}{5}-\frac{1}{2}\right)\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Least common multiple of 5 and 2 is 10. Convert \frac{7}{5} and \frac{1}{2} to fractions with denominator 10.

$$\sqrt{\frac{\left(\frac{14}{10}-\frac{5}{10}\right)\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Since \frac{14}{10} and \frac{5}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{14-5}{10}\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Subtract 5 from 14 to get 9.

$$\sqrt{\frac{\frac{9}{10}\times \frac{10}{9}+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Cancel out \frac{9}{10} and its reciprocal \frac{10}{9}.

$$\sqrt{\frac{1+2}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Add 1 and 2 to get 3.

$$\sqrt{\frac{3}{\frac{10}{3}\times \frac{6}{5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Multiply \frac{10}{3} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{3}{\frac{10\times 6}{3\times 5}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Do the multiplications in the fraction \frac{10\times 6}{3\times 5}.

$$\sqrt{\frac{3}{\frac{60}{15}\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Divide 60 by 15 to get 4.

$$\sqrt{\frac{3}{4\left(\frac{7}{4}\times \frac{1}{7}+\frac{1}{4}\right)}}$$

Multiply \frac{7}{4} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{3}{4\left(\frac{7\times 1}{4\times 7}+\frac{1}{4}\right)}}$$

Cancel out 7 in both numerator and denominator.

$$\sqrt{\frac{3}{4\left(\frac{1}{4}+\frac{1}{4}\right)}}$$

Since \frac{1}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{3}{4\times \frac{1+1}{4}}}$$

Add 1 and 1 to get 2.

$$\sqrt{\frac{3}{4\times \frac{2}{4}}}$$

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

$$\sqrt{\frac{3}{4\times \frac{1}{2}}}$$

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

$$\sqrt{\frac{3}{\frac{4}{2}}}$$

Divide 4 by 2 to get 2.

$$\sqrt{\frac{3}{2}}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\sqrt{3}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}\sqrt{2}}{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{2}$$