Solve sqrt{(1-7/12)*(1-(9/5-1/4)*(1-16/31)+(frac{5}{8}+frac{2}{3}):frac{31}{4}]}

$$\sqrt { ( 1 - \frac { 7 } { 12 } ) \times ( 1 - ( \frac { 9 } { 5 } - \frac { 1 } { 4 } ) \times ( 1 - \frac { 16 } { 31 } ) + ( \frac { 5 } { 8 } + \frac { 2 } { 3 } ) : \frac { 31 } { 4 } ] }$$

Evaluate

$$\frac{5}{12}\approx 0.416666667$$

Solution Steps

Convert 1 to fraction \frac{12}{12}.

$$\sqrt{\left(\frac{12}{12}-\frac{7}{12}\right)\left(1-\left(\frac{9}{5}-\frac{1}{4}\right)\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Since \frac{12}{12} and \frac{7}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{12-7}{12}\left(1-\left(\frac{9}{5}-\frac{1}{4}\right)\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Subtract 7 from 12 to get 5.

$$\sqrt{\frac{5}{12}\left(1-\left(\frac{9}{5}-\frac{1}{4}\right)\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Least common multiple of 5 and 4 is 20. Convert \frac{9}{5} and \frac{1}{4} to fractions with denominator 20.

$$\sqrt{\frac{5}{12}\left(1-\left(\frac{36}{20}-\frac{5}{20}\right)\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Since \frac{36}{20} and \frac{5}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5}{12}\left(1-\frac{36-5}{20}\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Subtract 5 from 36 to get 31.

$$\sqrt{\frac{5}{12}\left(1-\frac{31}{20}\left(1-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Convert 1 to fraction \frac{31}{31}.

$$\sqrt{\frac{5}{12}\left(1-\frac{31}{20}\left(\frac{31}{31}-\frac{16}{31}\right)+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Since \frac{31}{31} and \frac{16}{31} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5}{12}\left(1-\frac{31}{20}\times \frac{31-16}{31}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Subtract 16 from 31 to get 15.

$$\sqrt{\frac{5}{12}\left(1-\frac{31}{20}\times \frac{15}{31}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Multiply \frac{31}{20} times \frac{15}{31} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{5}{12}\left(1-\frac{31\times 15}{20\times 31}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Cancel out 31 in both numerator and denominator.

$$\sqrt{\frac{5}{12}\left(1-\frac{15}{20}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Reduce the fraction \frac{15}{20} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{5}{12}\left(1-\frac{3}{4}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Convert 1 to fraction \frac{4}{4}.

$$\sqrt{\frac{5}{12}\left(\frac{4}{4}-\frac{3}{4}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Since \frac{4}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5}{12}\left(\frac{4-3}{4}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Subtract 3 from 4 to get 1.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{\frac{5}{8}+\frac{2}{3}}{\frac{31}{4}}\right)}$$

Least common multiple of 8 and 3 is 24. Convert \frac{5}{8} and \frac{2}{3} to fractions with denominator 24.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{\frac{15}{24}+\frac{16}{24}}{\frac{31}{4}}\right)}$$

Since \frac{15}{24} and \frac{16}{24} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{\frac{15+16}{24}}{\frac{31}{4}}\right)}$$

Add 15 and 16 to get 31.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{\frac{31}{24}}{\frac{31}{4}}\right)}$$

Divide \frac{31}{24} by \frac{31}{4} by multiplying \frac{31}{24} by the reciprocal of \frac{31}{4}.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{31}{24}\times \frac{4}{31}\right)}$$

Multiply \frac{31}{24} times \frac{4}{31} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{31\times 4}{24\times 31}\right)}$$

Cancel out 31 in both numerator and denominator.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{4}{24}\right)}$$

Reduce the fraction \frac{4}{24} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{5}{12}\left(\frac{1}{4}+\frac{1}{6}\right)}$$

Least common multiple of 4 and 6 is 12. Convert \frac{1}{4} and \frac{1}{6} to fractions with denominator 12.

$$\sqrt{\frac{5}{12}\left(\frac{3}{12}+\frac{2}{12}\right)}$$

Since \frac{3}{12} and \frac{2}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5}{12}\times \frac{3+2}{12}}$$

Add 3 and 2 to get 5.

$$\sqrt{\frac{5}{12}\times \frac{5}{12}}$$

Multiply \frac{5}{12} times \frac{5}{12} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{5\times 5}{12\times 12}}$$

Do the multiplications in the fraction \frac{5\times 5}{12\times 12}.

$$\sqrt{\frac{25}{144}}$$

Rewrite the square root of the division \frac{25}{144} as the division of square roots \frac{\sqrt{25}}{\sqrt{144}}. Take the square root of both numerator and denominator.

$$\frac{5}{12}$$

Factor

$$\frac{5}{2 ^ {2} \cdot 3} = 0.4166666666666667$$