Solve sqrt{(12/11-19/33+2/3):(8/3-frac{7}{22}+frac{3}{21}-frac{2}{33})*(1-frac{7}{13})}

$$\sqrt { ( \frac { 12 } { 11 } - \frac { 19 } { 33 } + \frac { 2 } { 3 } ) : ( \frac { 8 } { 3 } - \frac { 7 } { 22 } + \frac { 3 } { 21 } - \frac { 2 } { 33 } ) \times ( 1 - \frac { 7 } { 13 } ) }$$

Evaluate

$$\frac{6\sqrt{7861}}{1123}\approx 0.473707643$$

Solution Steps

Least common multiple of 11 and 33 is 33. Convert \frac{12}{11} and \frac{19}{33} to fractions with denominator 33.

$$\sqrt{\frac{\frac{36}{33}-\frac{19}{33}+\frac{2}{3}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Since \frac{36}{33} and \frac{19}{33} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{36-19}{33}+\frac{2}{3}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Subtract 19 from 36 to get 17.

$$\sqrt{\frac{\frac{17}{33}+\frac{2}{3}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Least common multiple of 33 and 3 is 33. Convert \frac{17}{33} and \frac{2}{3} to fractions with denominator 33.

$$\sqrt{\frac{\frac{17}{33}+\frac{22}{33}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Since \frac{17}{33} and \frac{22}{33} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{17+22}{33}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Add 17 and 22 to get 39.

$$\sqrt{\frac{\frac{39}{33}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Reduce the fraction \frac{39}{33} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{\frac{13}{11}}{\frac{8}{3}-\frac{7}{22}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Least common multiple of 3 and 22 is 66. Convert \frac{8}{3} and \frac{7}{22} to fractions with denominator 66.

$$\sqrt{\frac{\frac{13}{11}}{\frac{176}{66}-\frac{21}{66}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Since \frac{176}{66} and \frac{21}{66} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{13}{11}}{\frac{176-21}{66}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Subtract 21 from 176 to get 155.

$$\sqrt{\frac{\frac{13}{11}}{\frac{155}{66}+\frac{3}{21}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Reduce the fraction \frac{3}{21} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{\frac{13}{11}}{\frac{155}{66}+\frac{1}{7}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Least common multiple of 66 and 7 is 462. Convert \frac{155}{66} and \frac{1}{7} to fractions with denominator 462.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1085}{462}+\frac{66}{462}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Since \frac{1085}{462} and \frac{66}{462} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1085+66}{462}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Add 1085 and 66 to get 1151.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1151}{462}-\frac{2}{33}}\left(1-\frac{7}{13}\right)}$$

Least common multiple of 462 and 33 is 462. Convert \frac{1151}{462} and \frac{2}{33} to fractions with denominator 462.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1151}{462}-\frac{28}{462}}\left(1-\frac{7}{13}\right)}$$

Since \frac{1151}{462} and \frac{28}{462} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1151-28}{462}}\left(1-\frac{7}{13}\right)}$$

Subtract 28 from 1151 to get 1123.

$$\sqrt{\frac{\frac{13}{11}}{\frac{1123}{462}}\left(1-\frac{7}{13}\right)}$$

Divide \frac{13}{11} by \frac{1123}{462} by multiplying \frac{13}{11} by the reciprocal of \frac{1123}{462}.

$$\sqrt{\frac{13}{11}\times \frac{462}{1123}\left(1-\frac{7}{13}\right)}$$

Multiply \frac{13}{11} times \frac{462}{1123} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{13\times 462}{11\times 1123}\left(1-\frac{7}{13}\right)}$$

Do the multiplications in the fraction \frac{13\times 462}{11\times 1123}.

$$\sqrt{\frac{6006}{12353}\left(1-\frac{7}{13}\right)}$$

Reduce the fraction \frac{6006}{12353} to lowest terms by extracting and canceling out 11.

$$\sqrt{\frac{546}{1123}\left(1-\frac{7}{13}\right)}$$

Convert 1 to fraction \frac{13}{13}.

$$\sqrt{\frac{546}{1123}\left(\frac{13}{13}-\frac{7}{13}\right)}$$

Since \frac{13}{13} and \frac{7}{13} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{546}{1123}\times \frac{13-7}{13}}$$

Subtract 7 from 13 to get 6.

$$\sqrt{\frac{546}{1123}\times \frac{6}{13}}$$

Multiply \frac{546}{1123} times \frac{6}{13} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{546\times 6}{1123\times 13}}$$

Do the multiplications in the fraction \frac{546\times 6}{1123\times 13}.

$$\sqrt{\frac{3276}{14599}}$$

Reduce the fraction \frac{3276}{14599} to lowest terms by extracting and canceling out 13.

$$\sqrt{\frac{252}{1123}}$$

Rewrite the square root of the division \sqrt{\frac{252}{1123}} as the division of square roots \frac{\sqrt{252}}{\sqrt{1123}}.

$$\frac{\sqrt{252}}{\sqrt{1123}}$$

Factor 252=6^{2}\times 7. Rewrite the square root of the product \sqrt{6^{2}\times 7} as the product of square roots \sqrt{6^{2}}\sqrt{7}. Take the square root of 6^{2}.

$$\frac{6\sqrt{7}}{\sqrt{1123}}$$

Rationalize the denominator of \frac{6\sqrt{7}}{\sqrt{1123}} by multiplying numerator and denominator by \sqrt{1123}.

$$\frac{6\sqrt{7}\sqrt{1123}}{\left(\sqrt{1123}\right)^{2}}$$

The square of \sqrt{1123} is 1123.

$$\frac{6\sqrt{7}\sqrt{1123}}{1123}$$

To multiply \sqrt{7} and \sqrt{1123}, multiply the numbers under the square root.

$$\frac{6\sqrt{7861}}{1123}$$