Solve sqrt{[3/4+1/12-(1/2-1/5)-frac{4}{9}]*[frac{11}{3}-(frac{1}{5}+frac{4}{15})]}

$$\sqrt { [ \frac { 3 } { 4 } + \frac { 1 } { 12 } - ( \frac { 1 } { 2 } - \frac { 1 } { 5 } ) - \frac { 4 } { 9 } ] \times [ \frac { 11 } { 3 } - ( \frac { 1 } { 5 } + \frac { 4 } { 15 } ) ] }$$

Evaluate

$$\frac{8}{15}\approx 0.533333333$$

Solution Steps

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

$$\sqrt{\left(\frac{9}{12}+\frac{1}{12}-\left(\frac{1}{2}-\frac{1}{5}\right)-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Since \frac{9}{12} and \frac{1}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{9+1}{12}-\left(\frac{1}{2}-\frac{1}{5}\right)-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Add 9 and 1 to get 10.

$$\sqrt{\left(\frac{10}{12}-\left(\frac{1}{2}-\frac{1}{5}\right)-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Reduce the fraction \frac{10}{12} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{5}{6}-\left(\frac{1}{2}-\frac{1}{5}\right)-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

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

$$\sqrt{\left(\frac{5}{6}-\left(\frac{5}{10}-\frac{2}{10}\right)-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Since \frac{5}{10} and \frac{2}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{5}{6}-\frac{5-2}{10}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Subtract 2 from 5 to get 3.

$$\sqrt{\left(\frac{5}{6}-\frac{3}{10}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Least common multiple of 6 and 10 is 30. Convert \frac{5}{6} and \frac{3}{10} to fractions with denominator 30.

$$\sqrt{\left(\frac{25}{30}-\frac{9}{30}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Since \frac{25}{30} and \frac{9}{30} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{25-9}{30}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Subtract 9 from 25 to get 16.

$$\sqrt{\left(\frac{16}{30}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Reduce the fraction \frac{16}{30} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{8}{15}-\frac{4}{9}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Least common multiple of 15 and 9 is 45. Convert \frac{8}{15} and \frac{4}{9} to fractions with denominator 45.

$$\sqrt{\left(\frac{24}{45}-\frac{20}{45}\right)\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Since \frac{24}{45} and \frac{20}{45} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{24-20}{45}\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Subtract 20 from 24 to get 4.

$$\sqrt{\frac{4}{45}\left(\frac{11}{3}-\left(\frac{1}{5}+\frac{4}{15}\right)\right)}$$

Least common multiple of 5 and 15 is 15. Convert \frac{1}{5} and \frac{4}{15} to fractions with denominator 15.

$$\sqrt{\frac{4}{45}\left(\frac{11}{3}-\left(\frac{3}{15}+\frac{4}{15}\right)\right)}$$

Since \frac{3}{15} and \frac{4}{15} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{4}{45}\left(\frac{11}{3}-\frac{3+4}{15}\right)}$$

Add 3 and 4 to get 7.

$$\sqrt{\frac{4}{45}\left(\frac{11}{3}-\frac{7}{15}\right)}$$

Least common multiple of 3 and 15 is 15. Convert \frac{11}{3} and \frac{7}{15} to fractions with denominator 15.

$$\sqrt{\frac{4}{45}\left(\frac{55}{15}-\frac{7}{15}\right)}$$

Since \frac{55}{15} and \frac{7}{15} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{4}{45}\times \frac{55-7}{15}}$$

Subtract 7 from 55 to get 48.

$$\sqrt{\frac{4}{45}\times \frac{48}{15}}$$

Reduce the fraction \frac{48}{15} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{4}{45}\times \frac{16}{5}}$$

Multiply \frac{4}{45} times \frac{16}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{4\times 16}{45\times 5}}$$

Do the multiplications in the fraction \frac{4\times 16}{45\times 5}.

$$\sqrt{\frac{64}{225}}$$

Rewrite the square root of the division \frac{64}{225} as the division of square roots \frac{\sqrt{64}}{\sqrt{225}}. Take the square root of both numerator and denominator.

$$\frac{8}{15}$$

Factor

$$\frac{2 ^ {3}}{3 \cdot 5} = 0.5333333333333333$$