Solve sqrt{{[(3/5+1/10):7/20-(6/5+frac{7}{2}-frac{14}{5})]:frac{2}{3}-frac{1}{15}}:(frac{2}{3})^2}

$$\sqrt { \{ [ ( \frac { 3 } { 5 } + \frac { 1 } { 10 } ) : \frac { 7 } { 20 } - ( \frac { 6 } { 5 } + \frac { 7 } { 2 } - \frac { 14 } { 5 } ) ] : \frac { 2 } { 3 } - \frac { 1 } { 15 } \} : ( \frac { 2 } { 3 } ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{3}}{4}\approx 0.433012702$$

Solution Steps

Least common multiple of 5 and 10 is 10. Convert \frac{3}{5} and \frac{1}{10} to fractions with denominator 10.

$$\sqrt{\frac{\frac{\frac{\frac{6}{10}+\frac{1}{10}}{\frac{7}{20}}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Since \frac{6}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{\frac{\frac{6+1}{10}}{\frac{7}{20}}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Add 6 and 1 to get 7.

$$\sqrt{\frac{\frac{\frac{\frac{7}{10}}{\frac{7}{20}}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Divide \frac{7}{10} by \frac{7}{20} by multiplying \frac{7}{10} by the reciprocal of \frac{7}{20}.

$$\sqrt{\frac{\frac{\frac{7}{10}\times \frac{20}{7}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Multiply \frac{7}{10} times \frac{20}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{\frac{7\times 20}{10\times 7}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Cancel out 7 in both numerator and denominator.

$$\sqrt{\frac{\frac{\frac{20}{10}-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Divide 20 by 10 to get 2.

$$\sqrt{\frac{\frac{2-\left(\frac{6}{5}+\frac{7}{2}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Least common multiple of 5 and 2 is 10. Convert \frac{6}{5} and \frac{7}{2} to fractions with denominator 10.

$$\sqrt{\frac{\frac{2-\left(\frac{12}{10}+\frac{35}{10}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Since \frac{12}{10} and \frac{35}{10} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{2-\left(\frac{12+35}{10}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Add 12 and 35 to get 47.

$$\sqrt{\frac{\frac{2-\left(\frac{47}{10}-\frac{14}{5}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Least common multiple of 10 and 5 is 10. Convert \frac{47}{10} and \frac{14}{5} to fractions with denominator 10.

$$\sqrt{\frac{\frac{2-\left(\frac{47}{10}-\frac{28}{10}\right)}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Since \frac{47}{10} and \frac{28}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{2-\frac{47-28}{10}}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Subtract 28 from 47 to get 19.

$$\sqrt{\frac{\frac{2-\frac{19}{10}}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Convert 2 to fraction \frac{20}{10}.

$$\sqrt{\frac{\frac{\frac{20}{10}-\frac{19}{10}}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Since \frac{20}{10} and \frac{19}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{\frac{20-19}{10}}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Subtract 19 from 20 to get 1.

$$\sqrt{\frac{\frac{\frac{1}{10}}{\frac{2}{3}}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Divide \frac{1}{10} by \frac{2}{3} by multiplying \frac{1}{10} by the reciprocal of \frac{2}{3}.

$$\sqrt{\frac{\frac{1}{10}\times \frac{3}{2}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Multiply \frac{1}{10} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{1\times 3}{10\times 2}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Do the multiplications in the fraction \frac{1\times 3}{10\times 2}.

$$\sqrt{\frac{\frac{3}{20}-\frac{1}{15}}{\left(\frac{2}{3}\right)^{2}}}$$

Least common multiple of 20 and 15 is 60. Convert \frac{3}{20} and \frac{1}{15} to fractions with denominator 60.

$$\sqrt{\frac{\frac{9}{60}-\frac{4}{60}}{\left(\frac{2}{3}\right)^{2}}}$$

Since \frac{9}{60} and \frac{4}{60} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{9-4}{60}}{\left(\frac{2}{3}\right)^{2}}}$$

Subtract 4 from 9 to get 5.

$$\sqrt{\frac{\frac{5}{60}}{\left(\frac{2}{3}\right)^{2}}}$$

Reduce the fraction \frac{5}{60} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{\frac{1}{12}}{\left(\frac{2}{3}\right)^{2}}}$$

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

$$\sqrt{\frac{\frac{1}{12}}{\frac{4}{9}}}$$

Divide \frac{1}{12} by \frac{4}{9} by multiplying \frac{1}{12} by the reciprocal of \frac{4}{9}.

$$\sqrt{\frac{1}{12}\times \frac{9}{4}}$$

Multiply \frac{1}{12} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1\times 9}{12\times 4}}$$

Do the multiplications in the fraction \frac{1\times 9}{12\times 4}.

$$\sqrt{\frac{9}{48}}$$

Reduce the fraction \frac{9}{48} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{3}{16}}$$

Rewrite the square root of the division \sqrt{\frac{3}{16}} as the division of square roots \frac{\sqrt{3}}{\sqrt{16}}.

$$\frac{\sqrt{3}}{\sqrt{16}}$$

Calculate the square root of 16 and get 4.

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