Solve sqrt{(7/2+5/3)-[(1/4+3-1/3-frac{13}{6})-(1-frac{5}{9})]-frac{31}{36}}

$$\sqrt { ( \frac { 7 } { 2 } + \frac { 5 } { 3 } ) - [ ( \frac { 1 } { 4 } + 3 - \frac { 1 } { 3 } - \frac { 13 } { 6 } ) - ( 1 - \frac { 5 } { 9 } ) ] - \frac { 31 } { 36 } }$$

Evaluate

$$2$$

Solution Steps

Least common multiple of 2 and 3 is 6. Convert \frac{7}{2} and \frac{5}{3} to fractions with denominator 6.

$$\sqrt{\frac{21}{6}+\frac{10}{6}-\left(\frac{1}{4}+3-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Since \frac{21}{6} and \frac{10}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{21+10}{6}-\left(\frac{1}{4}+3-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Add 21 and 10 to get 31.

$$\sqrt{\frac{31}{6}-\left(\frac{1}{4}+3-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Convert 3 to fraction \frac{12}{4}.

$$\sqrt{\frac{31}{6}-\left(\frac{1}{4}+\frac{12}{4}-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Since \frac{1}{4} and \frac{12}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{31}{6}-\left(\frac{1+12}{4}-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Add 1 and 12 to get 13.

$$\sqrt{\frac{31}{6}-\left(\frac{13}{4}-\frac{1}{3}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Least common multiple of 4 and 3 is 12. Convert \frac{13}{4} and \frac{1}{3} to fractions with denominator 12.

$$\sqrt{\frac{31}{6}-\left(\frac{39}{12}-\frac{4}{12}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Since \frac{39}{12} and \frac{4}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{31}{6}-\left(\frac{39-4}{12}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Subtract 4 from 39 to get 35.

$$\sqrt{\frac{31}{6}-\left(\frac{35}{12}-\frac{13}{6}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Least common multiple of 12 and 6 is 12. Convert \frac{35}{12} and \frac{13}{6} to fractions with denominator 12.

$$\sqrt{\frac{31}{6}-\left(\frac{35}{12}-\frac{26}{12}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Since \frac{35}{12} and \frac{26}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{31}{6}-\left(\frac{35-26}{12}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Subtract 26 from 35 to get 9.

$$\sqrt{\frac{31}{6}-\left(\frac{9}{12}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Reduce the fraction \frac{9}{12} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{31}{6}-\left(\frac{3}{4}-\left(1-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Convert 1 to fraction \frac{9}{9}.

$$\sqrt{\frac{31}{6}-\left(\frac{3}{4}-\left(\frac{9}{9}-\frac{5}{9}\right)\right)-\frac{31}{36}}$$

Since \frac{9}{9} and \frac{5}{9} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{31}{6}-\left(\frac{3}{4}-\frac{9-5}{9}\right)-\frac{31}{36}}$$

Subtract 5 from 9 to get 4.

$$\sqrt{\frac{31}{6}-\left(\frac{3}{4}-\frac{4}{9}\right)-\frac{31}{36}}$$

Least common multiple of 4 and 9 is 36. Convert \frac{3}{4} and \frac{4}{9} to fractions with denominator 36.

$$\sqrt{\frac{31}{6}-\left(\frac{27}{36}-\frac{16}{36}\right)-\frac{31}{36}}$$

Since \frac{27}{36} and \frac{16}{36} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{31}{6}-\frac{27-16}{36}-\frac{31}{36}}$$

Subtract 16 from 27 to get 11.

$$\sqrt{\frac{31}{6}-\frac{11}{36}-\frac{31}{36}}$$

Least common multiple of 6 and 36 is 36. Convert \frac{31}{6} and \frac{11}{36} to fractions with denominator 36.

$$\sqrt{\frac{186}{36}-\frac{11}{36}-\frac{31}{36}}$$

Since \frac{186}{36} and \frac{11}{36} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{186-11}{36}-\frac{31}{36}}$$

Subtract 11 from 186 to get 175.

$$\sqrt{\frac{175}{36}-\frac{31}{36}}$$

Since \frac{175}{36} and \frac{31}{36} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{175-31}{36}}$$

Subtract 31 from 175 to get 144.

$$\sqrt{\frac{144}{36}}$$

Divide 144 by 36 to get 4.

$$\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$2$$

Factor

$$2$$