Solve 155sqrt{{5/7*[1/4+(7/5-3/4):(frac{3}{2}+5)]+frac{3}{4}}:[frac{3}{22}*(2-frac{16}{99})*frac{3}{2}-frac{1}{3}:(frac{11}{6})^2-frac{17}{11}*frac{1}{22}]}

$$155 \sqrt { \{ \frac { 5 } { 7 } \times [ \frac { 1 } { 4 } + ( \frac { 7 } { 5 } - \frac { 3 } { 4 } ) : ( \frac { 3 } { 2 } + 5 ) ] + \frac { 3 } { 4 } \} : [ \frac { 3 } { 22 } \times ( 2 - \frac { 16 } { 99 } ) \times \frac { 3 } { 2 } - \frac { 1 } { 3 } : ( \frac { 11 } { 6 } ) ^ { 2 } - \frac { 17 } { 11 } \times \frac { 1 } { 22 } ] }$$

Evaluate

$$341$$

Solution Steps

Least common multiple of 5 and 4 is 20. Convert \frac{7}{5} and \frac{3}{4} to fractions with denominator 20.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{28}{20}-\frac{15}{20}}{\frac{3}{2}+5}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{28}{20} and \frac{15}{20} have the same denominator, subtract them by subtracting their numerators.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{28-15}{20}}{\frac{3}{2}+5}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Subtract 15 from 28 to get 13.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{13}{20}}{\frac{3}{2}+5}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Convert 5 to fraction \frac{10}{2}.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{13}{20}}{\frac{3}{2}+\frac{10}{2}}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{3}{2} and \frac{10}{2} have the same denominator, add them by adding their numerators.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{13}{20}}{\frac{3+10}{2}}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Add 3 and 10 to get 13.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{\frac{13}{20}}{\frac{13}{2}}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Divide \frac{13}{20} by \frac{13}{2} by multiplying \frac{13}{20} by the reciprocal of \frac{13}{2}.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{13}{20}\times \frac{2}{13}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{13}{20} times \frac{2}{13} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{13\times 2}{20\times 13}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Cancel out 13 in both numerator and denominator.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{2}{20}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Reduce the fraction \frac{2}{20} to lowest terms by extracting and canceling out 2.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{1}{4}+\frac{1}{10}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Least common multiple of 4 and 10 is 20. Convert \frac{1}{4} and \frac{1}{10} to fractions with denominator 20.

$$155\sqrt{\frac{\frac{5}{7}\left(\frac{5}{20}+\frac{2}{20}\right)+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{5}{20} and \frac{2}{20} have the same denominator, add them by adding their numerators.

$$155\sqrt{\frac{\frac{5}{7}\times \frac{5+2}{20}+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Add 5 and 2 to get 7.

$$155\sqrt{\frac{\frac{5}{7}\times \frac{7}{20}+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{5}{7} times \frac{7}{20} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{\frac{5\times 7}{7\times 20}+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Cancel out 7 in both numerator and denominator.

$$155\sqrt{\frac{\frac{5}{20}+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Reduce the fraction \frac{5}{20} to lowest terms by extracting and canceling out 5.

$$155\sqrt{\frac{\frac{1}{4}+\frac{3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{1}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$155\sqrt{\frac{\frac{1+3}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Add 1 and 3 to get 4.

$$155\sqrt{\frac{\frac{4}{4}}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Divide 4 by 4 to get 1.

$$155\sqrt{\frac{1}{\frac{3}{22}\left(2-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Convert 2 to fraction \frac{198}{99}.

$$155\sqrt{\frac{1}{\frac{3}{22}\left(\frac{198}{99}-\frac{16}{99}\right)\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{198}{99} and \frac{16}{99} have the same denominator, subtract them by subtracting their numerators.

$$155\sqrt{\frac{1}{\frac{3}{22}\times \frac{198-16}{99}\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Subtract 16 from 198 to get 182.

$$155\sqrt{\frac{1}{\frac{3}{22}\times \frac{182}{99}\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{3}{22} times \frac{182}{99} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{1}{\frac{3\times 182}{22\times 99}\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Do the multiplications in the fraction \frac{3\times 182}{22\times 99}.

$$155\sqrt{\frac{1}{\frac{546}{2178}\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Reduce the fraction \frac{546}{2178} to lowest terms by extracting and canceling out 6.

$$155\sqrt{\frac{1}{\frac{91}{363}\times \frac{3}{2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{91}{363} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{1}{\frac{91\times 3}{363\times 2}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Do the multiplications in the fraction \frac{91\times 3}{363\times 2}.

$$155\sqrt{\frac{1}{\frac{273}{726}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Reduce the fraction \frac{273}{726} to lowest terms by extracting and canceling out 3.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{\frac{1}{3}}{\left(\frac{11}{6}\right)^{2}}-\frac{17}{11}\times \frac{1}{22}}}$$

Calculate \frac{11}{6} to the power of 2 and get \frac{121}{36}.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{\frac{1}{3}}{\frac{121}{36}}-\frac{17}{11}\times \frac{1}{22}}}$$

Divide \frac{1}{3} by \frac{121}{36} by multiplying \frac{1}{3} by the reciprocal of \frac{121}{36}.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{1}{3}\times \frac{36}{121}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{1}{3} times \frac{36}{121} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{1\times 36}{3\times 121}-\frac{17}{11}\times \frac{1}{22}}}$$

Do the multiplications in the fraction \frac{1\times 36}{3\times 121}.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{36}{363}-\frac{17}{11}\times \frac{1}{22}}}$$

Reduce the fraction \frac{36}{363} to lowest terms by extracting and canceling out 3.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{12}{121}-\frac{17}{11}\times \frac{1}{22}}}$$

Least common multiple of 242 and 121 is 242. Convert \frac{91}{242} and \frac{12}{121} to fractions with denominator 242.

$$155\sqrt{\frac{1}{\frac{91}{242}-\frac{24}{242}-\frac{17}{11}\times \frac{1}{22}}}$$

Since \frac{91}{242} and \frac{24}{242} have the same denominator, subtract them by subtracting their numerators.

$$155\sqrt{\frac{1}{\frac{91-24}{242}-\frac{17}{11}\times \frac{1}{22}}}$$

Subtract 24 from 91 to get 67.

$$155\sqrt{\frac{1}{\frac{67}{242}-\frac{17}{11}\times \frac{1}{22}}}$$

Multiply \frac{17}{11} times \frac{1}{22} by multiplying numerator times numerator and denominator times denominator.

$$155\sqrt{\frac{1}{\frac{67}{242}-\frac{17\times 1}{11\times 22}}}$$

Do the multiplications in the fraction \frac{17\times 1}{11\times 22}.

$$155\sqrt{\frac{1}{\frac{67}{242}-\frac{17}{242}}}$$

Since \frac{67}{242} and \frac{17}{242} have the same denominator, subtract them by subtracting their numerators.

$$155\sqrt{\frac{1}{\frac{67-17}{242}}}$$

Subtract 17 from 67 to get 50.

$$155\sqrt{\frac{1}{\frac{50}{242}}}$$

Reduce the fraction \frac{50}{242} to lowest terms by extracting and canceling out 2.

$$155\sqrt{\frac{1}{\frac{25}{121}}}$$

Divide 1 by \frac{25}{121} by multiplying 1 by the reciprocal of \frac{25}{121}.

$$155\sqrt{1\times \frac{121}{25}}$$

Multiply 1 and \frac{121}{25} to get \frac{121}{25}.

$$155\sqrt{\frac{121}{25}}$$

Rewrite the square root of the division \frac{121}{25} as the division of square roots \frac{\sqrt{121}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$155\times \frac{11}{5}$$

Express 155\times \frac{11}{5} as a single fraction.

$$\frac{155\times 11}{5}$$

Multiply 155 and 11 to get 1705.

$$\frac{1705}{5}$$

Divide 1705 by 5 to get 341.

$$341$$

Factor

$$11\times 31$$