Solve sqrt{13/6-35/66+27/1215/3-left(frac{14}{15}+frac{8}{165}right)left(frac{2}{9}+frac{11}{18}right)}

$$\sqrt{ \frac{ 13 }{ 6 } - \frac{ 35 }{ 66 } + \frac{ 27 }{ 121 } \frac{ 5 }{ 3 } - \left( \frac{ 14 }{ 15 } + \frac{ 8 }{ 165 } \right) \left( \frac{ 2 }{ 9 } + \frac{ 11 }{ 18 } \right) }$$

Evaluate

$$\frac{12}{11}\approx 1.090909091$$

Solution Steps

Least common multiple of 6 and 66 is 66. Convert \frac{13}{6} and \frac{35}{66} to fractions with denominator 66.

$$\sqrt{\frac{143}{66}-\frac{35}{66}+\frac{27}{121}\times \frac{5}{3}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Since \frac{143}{66} and \frac{35}{66} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{143-35}{66}+\frac{27}{121}\times \frac{5}{3}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Subtract 35 from 143 to get 108.

$$\sqrt{\frac{108}{66}+\frac{27}{121}\times \frac{5}{3}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Reduce the fraction \frac{108}{66} to lowest terms by extracting and canceling out 6.

$$\sqrt{\frac{18}{11}+\frac{27}{121}\times \frac{5}{3}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Multiply \frac{27}{121} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{18}{11}+\frac{27\times 5}{121\times 3}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Do the multiplications in the fraction \frac{27\times 5}{121\times 3}.

$$\sqrt{\frac{18}{11}+\frac{135}{363}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Reduce the fraction \frac{135}{363} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{18}{11}+\frac{45}{121}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Least common multiple of 11 and 121 is 121. Convert \frac{18}{11} and \frac{45}{121} to fractions with denominator 121.

$$\sqrt{\frac{198}{121}+\frac{45}{121}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Since \frac{198}{121} and \frac{45}{121} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{198+45}{121}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Add 198 and 45 to get 243.

$$\sqrt{\frac{243}{121}-\left(\frac{14}{15}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Least common multiple of 15 and 165 is 165. Convert \frac{14}{15} and \frac{8}{165} to fractions with denominator 165.

$$\sqrt{\frac{243}{121}-\left(\frac{154}{165}+\frac{8}{165}\right)\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Since \frac{154}{165} and \frac{8}{165} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{243}{121}-\frac{154+8}{165}\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Add 154 and 8 to get 162.

$$\sqrt{\frac{243}{121}-\frac{162}{165}\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Reduce the fraction \frac{162}{165} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{243}{121}-\frac{54}{55}\left(\frac{2}{9}+\frac{11}{18}\right)}$$

Least common multiple of 9 and 18 is 18. Convert \frac{2}{9} and \frac{11}{18} to fractions with denominator 18.

$$\sqrt{\frac{243}{121}-\frac{54}{55}\left(\frac{4}{18}+\frac{11}{18}\right)}$$

Since \frac{4}{18} and \frac{11}{18} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{243}{121}-\frac{54}{55}\times \frac{4+11}{18}}$$

Add 4 and 11 to get 15.

$$\sqrt{\frac{243}{121}-\frac{54}{55}\times \frac{15}{18}}$$

Reduce the fraction \frac{15}{18} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{243}{121}-\frac{54}{55}\times \frac{5}{6}}$$

Multiply \frac{54}{55} times \frac{5}{6} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{243}{121}-\frac{54\times 5}{55\times 6}}$$

Do the multiplications in the fraction \frac{54\times 5}{55\times 6}.

$$\sqrt{\frac{243}{121}-\frac{270}{330}}$$

Reduce the fraction \frac{270}{330} to lowest terms by extracting and canceling out 30.

$$\sqrt{\frac{243}{121}-\frac{9}{11}}$$

Least common multiple of 121 and 11 is 121. Convert \frac{243}{121} and \frac{9}{11} to fractions with denominator 121.

$$\sqrt{\frac{243}{121}-\frac{99}{121}}$$

Since \frac{243}{121} and \frac{99}{121} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{243-99}{121}}$$

Subtract 99 from 243 to get 144.

$$\sqrt{\frac{144}{121}}$$

Rewrite the square root of the division \frac{144}{121} as the division of square roots \frac{\sqrt{144}}{\sqrt{121}}. Take the square root of both numerator and denominator.

$$\frac{12}{11}$$

Factor

$$\frac{2 ^ {2} \cdot 3}{11} = 1\frac{1}{11} = 1.0909090909090908$$