Solve {l}{sqrt{1+[19/9+(7/3+3/4-1/12)-(5-frac{1}{2}:frac{3}{4})]}-sqrt{(frac{1}{3}+frac{1}{9}+5*frac{1}{15})*(frac{2^2}{3}-frac{5}{3^2})}}{[(frac{1}{1}+frac{5}{2})*frac{9}{9}+(2-frac{1}{1})]+35}

$$\left. \begin{array} { l } { \sqrt { 1 + [ \frac { 19 } { 9 } + ( \frac { 7 } { 3 } + \frac { 3 } { 4 } - \frac { 1 } { 12 } ) - ( 5 - \frac { 1 } { 2 } : \frac { 3 } { 4 } ) ] } - \sqrt { ( \frac { 1 } { 3 } + \frac { 1 } { 9 } + 5 \times \frac { 1 } { 15 } ) \times ( \frac { 2 ^ { 2 } } { 3 } - \frac { 5 } { 3 ^ { 2 } } ) } } \\ { [ ( \frac { 1 } { 1 } + \frac { 5 } { 2 } ) \cdot \frac { 9 } { 9 } + ( 2 - \frac { 1 } { 1 } ) ] + 35 } \end{array} \right.$$

Sort

$$\frac{5}{9},\ \frac{79}{2}$$

Solution Steps

Divide 1 by 1 to get 1.

$$sort(\sqrt{1+\frac{19}{9}+\frac{7}{3}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times \frac{9}{9}+2-\frac{1}{1}+35)$$

Divide 9 by 9 to get 1.

$$sort(\sqrt{1+\frac{19}{9}+\frac{7}{3}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-\frac{1}{1}+35)$$

Divide 1 by 1 to get 1.

$$sort(\sqrt{1+\frac{19}{9}+\frac{7}{3}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 9 and 3 is 9. Convert \frac{19}{9} and \frac{7}{3} to fractions with denominator 9.

$$sort(\sqrt{1+\frac{19}{9}+\frac{21}{9}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{19}{9} and \frac{21}{9} have the same denominator, add them by adding their numerators.

$$sort(\sqrt{1+\frac{19+21}{9}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Add 19 and 21 to get 40.

$$sort(\sqrt{1+\frac{40}{9}+\frac{3}{4}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 9 and 4 is 36. Convert \frac{40}{9} and \frac{3}{4} to fractions with denominator 36.

$$sort(\sqrt{1+\frac{160}{36}+\frac{27}{36}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{160}{36} and \frac{27}{36} have the same denominator, add them by adding their numerators.

$$sort(\sqrt{1+\frac{160+27}{36}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Add 160 and 27 to get 187.

$$sort(\sqrt{1+\frac{187}{36}-\frac{1}{12}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 36 and 12 is 36. Convert \frac{187}{36} and \frac{1}{12} to fractions with denominator 36.

$$sort(\sqrt{1+\frac{187}{36}-\frac{3}{36}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{187}{36} and \frac{3}{36} have the same denominator, subtract them by subtracting their numerators.

$$sort(\sqrt{1+\frac{187-3}{36}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Subtract 3 from 187 to get 184.

$$sort(\sqrt{1+\frac{184}{36}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Reduce the fraction \frac{184}{36} to lowest terms by extracting and canceling out 4.

$$sort(\sqrt{1+\frac{46}{9}-\left(5-\frac{\frac{1}{2}}{\frac{3}{4}}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Divide \frac{1}{2} by \frac{3}{4} by multiplying \frac{1}{2} by the reciprocal of \frac{3}{4}.

$$sort(\sqrt{1+\frac{46}{9}-\left(5-\frac{1}{2}\times \frac{4}{3}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Multiply \frac{1}{2} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

$$sort(\sqrt{1+\frac{46}{9}-\left(5-\frac{1\times 4}{2\times 3}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Do the multiplications in the fraction \frac{1\times 4}{2\times 3}.

$$sort(\sqrt{1+\frac{46}{9}-\left(5-\frac{4}{6}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$sort(\sqrt{1+\frac{46}{9}-\left(5-\frac{2}{3}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Convert 5 to fraction \frac{15}{3}.

$$sort(\sqrt{1+\frac{46}{9}-\left(\frac{15}{3}-\frac{2}{3}\right)}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{15}{3} and \frac{2}{3} have the same denominator, subtract them by subtracting their numerators.

$$sort(\sqrt{1+\frac{46}{9}-\frac{15-2}{3}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Subtract 2 from 15 to get 13.

$$sort(\sqrt{1+\frac{46}{9}-\frac{13}{3}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 9 and 3 is 9. Convert \frac{46}{9} and \frac{13}{3} to fractions with denominator 9.

$$sort(\sqrt{1+\frac{46}{9}-\frac{39}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{46}{9} and \frac{39}{9} have the same denominator, subtract them by subtracting their numerators.

$$sort(\sqrt{1+\frac{46-39}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Subtract 39 from 46 to get 7.

$$sort(\sqrt{1+\frac{7}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Convert 1 to fraction \frac{9}{9}.

$$sort(\sqrt{\frac{9}{9}+\frac{7}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{9}{9} and \frac{7}{9} have the same denominator, add them by adding their numerators.

$$sort(\sqrt{\frac{9+7}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Add 9 and 7 to get 16.

$$sort(\sqrt{\frac{16}{9}}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Rewrite the square root of the division \frac{16}{9} as the division of square roots \frac{\sqrt{16}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{1}{3}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 3 and 9 is 9. Convert \frac{1}{3} and \frac{1}{9} to fractions with denominator 9.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{3}{9}+\frac{1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{3}{9} and \frac{1}{9} have the same denominator, add them by adding their numerators.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{3+1}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Add 3 and 1 to get 4.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{4}{9}+5\times \frac{1}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Multiply 5 and \frac{1}{15} to get \frac{5}{15}.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{4}{9}+\frac{5}{15}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Reduce the fraction \frac{5}{15} to lowest terms by extracting and canceling out 5.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{4}{9}+\frac{1}{3}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 9 and 3 is 9. Convert \frac{4}{9} and \frac{1}{3} to fractions with denominator 9.

$$sort(\frac{4}{3}-\sqrt{\left(\frac{4}{9}+\frac{3}{9}\right)\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{4}{9} and \frac{3}{9} have the same denominator, add them by adding their numerators.

$$sort(\frac{4}{3}-\sqrt{\frac{4+3}{9}\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Add 4 and 3 to get 7.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\left(\frac{2^{2}}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Calculate 2 to the power of 2 and get 4.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\left(\frac{4}{3}-\frac{5}{3^{2}}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Calculate 3 to the power of 2 and get 9.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\left(\frac{4}{3}-\frac{5}{9}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 3 and 9 is 9. Convert \frac{4}{3} and \frac{5}{9} to fractions with denominator 9.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\left(\frac{12}{9}-\frac{5}{9}\right)},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{12}{9} and \frac{5}{9} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\times \frac{12-5}{9}},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Subtract 5 from 12 to get 7.

$$sort(\frac{4}{3}-\sqrt{\frac{7}{9}\times \frac{7}{9}},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Multiply \frac{7}{9} times \frac{7}{9} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{4}{3}-\sqrt{\frac{7\times 7}{9\times 9}},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Do the multiplications in the fraction \frac{7\times 7}{9\times 9}.

$$sort(\frac{4}{3}-\sqrt{\frac{49}{81}},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Rewrite the square root of the division \frac{49}{81} as the division of square roots \frac{\sqrt{49}}{\sqrt{81}}. Take the square root of both numerator and denominator.

$$sort(\frac{4}{3}-\frac{7}{9},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Least common multiple of 3 and 9 is 9. Convert \frac{4}{3} and \frac{7}{9} to fractions with denominator 9.

$$sort(\frac{12}{9}-\frac{7}{9},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Since \frac{12}{9} and \frac{7}{9} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{12-7}{9},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Subtract 7 from 12 to get 5.

$$sort(\frac{5}{9},\left(1+\frac{5}{2}\right)\times 1+2-1+35)$$

Convert 1 to fraction \frac{2}{2}.

$$sort(\frac{5}{9},\left(\frac{2}{2}+\frac{5}{2}\right)\times 1+2-1+35)$$

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

$$sort(\frac{5}{9},\frac{2+5}{2}\times 1+2-1+35)$$

Add 2 and 5 to get 7.

$$sort(\frac{5}{9},\frac{7}{2}\times 1+2-1+35)$$

Multiply \frac{7}{2} and 1 to get \frac{7}{2}.

$$sort(\frac{5}{9},\frac{7}{2}+2-1+35)$$

Convert 2 to fraction \frac{4}{2}.

$$sort(\frac{5}{9},\frac{7}{2}+\frac{4}{2}-1+35)$$

Since \frac{7}{2} and \frac{4}{2} have the same denominator, add them by adding their numerators.

$$sort(\frac{5}{9},\frac{7+4}{2}-1+35)$$

Add 7 and 4 to get 11.

$$sort(\frac{5}{9},\frac{11}{2}-1+35)$$

Convert 1 to fraction \frac{2}{2}.

$$sort(\frac{5}{9},\frac{11}{2}-\frac{2}{2}+35)$$

Since \frac{11}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{5}{9},\frac{11-2}{2}+35)$$

Subtract 2 from 11 to get 9.

$$sort(\frac{5}{9},\frac{9}{2}+35)$$

Convert 35 to fraction \frac{70}{2}.

$$sort(\frac{5}{9},\frac{9}{2}+\frac{70}{2})$$

Since \frac{9}{2} and \frac{70}{2} have the same denominator, add them by adding their numerators.

$$sort(\frac{5}{9},\frac{9+70}{2})$$

Add 9 and 70 to get 79.

$$sort(\frac{5}{9},\frac{79}{2})$$

Least common denominator of the numbers in the list \frac{5}{9},\frac{79}{2} is 18. Convert numbers in the list to fractions with denominator 18.

$$\frac{10}{18},\frac{711}{18}$$

Evaluate

$$\frac{5}{9},\ \frac{79}{2}$$