Solve sqrt{2/18*{[11/3-(5/6-3/4)]:[(3-frac{11}{4})+frac{10}{3}]}}

$$\sqrt { \frac { 2 } { 18 } \times \{ [ \frac { 11 } { 3 } - ( \frac { 5 } { 6 } - \frac { 3 } { 4 } ) ] : [ ( 3 - \frac { 11 } { 4 } ) + \frac { 10 } { 3 } ] \} }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

Reduce the fraction \frac{2}{18} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1}{9}\times \frac{\frac{11}{3}-\left(\frac{5}{6}-\frac{3}{4}\right)}{3-\frac{11}{4}+\frac{10}{3}}}$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{3}{4} to fractions with denominator 12.

$$\sqrt{\frac{1}{9}\times \frac{\frac{11}{3}-\left(\frac{10}{12}-\frac{9}{12}\right)}{3-\frac{11}{4}+\frac{10}{3}}}$$

Since \frac{10}{12} and \frac{9}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{9}\times \frac{\frac{11}{3}-\frac{10-9}{12}}{3-\frac{11}{4}+\frac{10}{3}}}$$

Subtract 9 from 10 to get 1.

$$\sqrt{\frac{1}{9}\times \frac{\frac{11}{3}-\frac{1}{12}}{3-\frac{11}{4}+\frac{10}{3}}}$$

Least common multiple of 3 and 12 is 12. Convert \frac{11}{3} and \frac{1}{12} to fractions with denominator 12.

$$\sqrt{\frac{1}{9}\times \frac{\frac{44}{12}-\frac{1}{12}}{3-\frac{11}{4}+\frac{10}{3}}}$$

Since \frac{44}{12} and \frac{1}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{9}\times \frac{\frac{44-1}{12}}{3-\frac{11}{4}+\frac{10}{3}}}$$

Subtract 1 from 44 to get 43.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{3-\frac{11}{4}+\frac{10}{3}}}$$

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

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{12}{4}-\frac{11}{4}+\frac{10}{3}}}$$

Since \frac{12}{4} and \frac{11}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{12-11}{4}+\frac{10}{3}}}$$

Subtract 11 from 12 to get 1.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{1}{4}+\frac{10}{3}}}$$

Least common multiple of 4 and 3 is 12. Convert \frac{1}{4} and \frac{10}{3} to fractions with denominator 12.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{3}{12}+\frac{40}{12}}}$$

Since \frac{3}{12} and \frac{40}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{3+40}{12}}}$$

Add 3 and 40 to get 43.

$$\sqrt{\frac{1}{9}\times \frac{\frac{43}{12}}{\frac{43}{12}}}$$

Divide \frac{43}{12} by \frac{43}{12} to get 1.

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

Multiply \frac{1}{9} and 1 to get \frac{1}{9}.

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

Rewrite the square root of the division \frac{1}{9} as the division of square roots \frac{\sqrt{1}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$