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

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

Evaluate

$$1$$

Solution Steps

Express 2\times \frac{5}{6} as a single fraction.

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

Multiply 2 and 5 to get 10.

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

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

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

Since \frac{1}{3} and \frac{5}{3} have the same denominator, add them by adding their numerators.

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

Add 1 and 5 to get 6.

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

Divide 6 by 3 to get 2.

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

Convert 2 to fraction \frac{12}{6}.

$$\sqrt{\frac{\frac{6}{5}\left(\frac{12}{6}-\frac{1}{6}\right)+\frac{\left(\frac{3}{2}\right)^{2}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

Since \frac{12}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{6}{5}\times \frac{12-1}{6}+\frac{\left(\frac{3}{2}\right)^{2}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

Subtract 1 from 12 to get 11.

$$\sqrt{\frac{\frac{6}{5}\times \frac{11}{6}+\frac{\left(\frac{3}{2}\right)^{2}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

Multiply \frac{6}{5} times \frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{6\times 11}{5\times 6}+\frac{\left(\frac{3}{2}\right)^{2}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

Cancel out 6 in both numerator and denominator.

$$\sqrt{\frac{\frac{11}{5}+\frac{\left(\frac{3}{2}\right)^{2}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

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

$$\sqrt{\frac{\frac{11}{5}+\frac{\frac{9}{4}\times \frac{2}{3}}{3}-\frac{7}{10}}{2}}$$

Multiply \frac{9}{4} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{11}{5}+\frac{\frac{9\times 2}{4\times 3}}{3}-\frac{7}{10}}{2}}$$

Do the multiplications in the fraction \frac{9\times 2}{4\times 3}.

$$\sqrt{\frac{\frac{11}{5}+\frac{\frac{18}{12}}{3}-\frac{7}{10}}{2}}$$

Reduce the fraction \frac{18}{12} to lowest terms by extracting and canceling out 6.

$$\sqrt{\frac{\frac{11}{5}+\frac{\frac{3}{2}}{3}-\frac{7}{10}}{2}}$$

Express \frac{\frac{3}{2}}{3} as a single fraction.

$$\sqrt{\frac{\frac{11}{5}+\frac{3}{2\times 3}-\frac{7}{10}}{2}}$$

Cancel out 3 in both numerator and denominator.

$$\sqrt{\frac{\frac{11}{5}+\frac{1}{2}-\frac{7}{10}}{2}}$$

Least common multiple of 5 and 2 is 10. Convert \frac{11}{5} and \frac{1}{2} to fractions with denominator 10.

$$\sqrt{\frac{\frac{22}{10}+\frac{5}{10}-\frac{7}{10}}{2}}$$

Since \frac{22}{10} and \frac{5}{10} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{22+5}{10}-\frac{7}{10}}{2}}$$

Add 22 and 5 to get 27.

$$\sqrt{\frac{\frac{27}{10}-\frac{7}{10}}{2}}$$

Since \frac{27}{10} and \frac{7}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{27-7}{10}}{2}}$$

Subtract 7 from 27 to get 20.

$$\sqrt{\frac{\frac{20}{10}}{2}}$$

Divide 20 by 10 to get 2.

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

Divide 2 by 2 to get 1.

$$\sqrt{1}$$

Calculate the square root of 1 and get 1.

$$1$$

Factor

$$1$$