Solve {l}{sqrt{11^{2}-9^{2}-3*5}sqrt{13^{2}-12^{2}-3^{2}}}{sqrt{[(1/2^2+1/2)*(1-3/2^2)+(2/3^2-frac{1}{6}+frac{1}{3})*frac{9}{7}+frac{1}{2^2}]*frac{1}{15}}}

$$\left. \begin{array} { l } { \sqrt { 11 ^ { 2 } - 9 ^ { 2 } - 3 \times 5 } \sqrt { 13 ^ { 2 } - 12 ^ { 2 } - 3 ^ { 2 } } } \\ { \sqrt { [ ( \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 2 } ) \times ( 1 - \frac { 3 } { 2 ^ { 2 } } ) + ( \frac { 2 } { 3 ^ { 2 } } - \frac { 1 } { 6 } + \frac { 1 } { 3 } ) \times \frac { 9 } { 7 } + \frac { 1 } { 2 ^ { 2 } } ] \times \frac { 1 } { 15 } } } \end{array} \right.$$

Sort

$$\frac{1}{4},20$$

Solution Steps

Calculate 11 to the power of 2 and get 121.

$$sort(\sqrt{121-9^{2}-3\times 5}\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 9 to the power of 2 and get 81.

$$sort(\sqrt{121-81-3\times 5}\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 81 from 121 to get 40.

$$sort(\sqrt{40-3\times 5}\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Multiply 3 and 5 to get 15.

$$sort(\sqrt{40-15}\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 15 from 40 to get 25.

$$sort(\sqrt{25}\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate the square root of 25 and get 5.

$$sort(5\sqrt{13^{2}-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 13 to the power of 2 and get 169.

$$sort(5\sqrt{169-12^{2}-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 12 to the power of 2 and get 144.

$$sort(5\sqrt{169-144-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 144 from 169 to get 25.

$$sort(5\sqrt{25-3^{2}},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 3 to the power of 2 and get 9.

$$sort(5\sqrt{25-9},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 9 from 25 to get 16.

$$sort(5\sqrt{16},\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate the square root of 16 and get 4.

$$sort(5\times 4,\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Multiply 5 and 4 to get 20.

$$sort(20,\sqrt{\left(\left(\frac{1}{2^{2}}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 2 to the power of 2 and get 4.

$$sort(20,\sqrt{\left(\left(\frac{1}{4}+\frac{1}{2}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Least common multiple of 4 and 2 is 4. Convert \frac{1}{4} and \frac{1}{2} to fractions with denominator 4.

$$sort(20,\sqrt{\left(\left(\frac{1}{4}+\frac{2}{4}\right)\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Since \frac{1}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

$$sort(20,\sqrt{\left(\frac{1+2}{4}\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Add 1 and 2 to get 3.

$$sort(20,\sqrt{\left(\frac{3}{4}\left(1-\frac{3}{2^{2}}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 2 to the power of 2 and get 4.

$$sort(20,\sqrt{\left(\frac{3}{4}\left(1-\frac{3}{4}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Convert 1 to fraction \frac{4}{4}.

$$sort(20,\sqrt{\left(\frac{3}{4}\left(\frac{4}{4}-\frac{3}{4}\right)+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Since \frac{4}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$sort(20,\sqrt{\left(\frac{3}{4}\times \frac{4-3}{4}+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 3 from 4 to get 1.

$$sort(20,\sqrt{\left(\frac{3}{4}\times \frac{1}{4}+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

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

$$sort(20,\sqrt{\left(\frac{3\times 1}{4\times 4}+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

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

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{2}{3^{2}}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 3 to the power of 2 and get 9.

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{2}{9}-\frac{1}{6}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Least common multiple of 9 and 6 is 18. Convert \frac{2}{9} and \frac{1}{6} to fractions with denominator 18.

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{4}{18}-\frac{3}{18}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Since \frac{4}{18} and \frac{3}{18} have the same denominator, subtract them by subtracting their numerators.

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{4-3}{18}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Subtract 3 from 4 to get 1.

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{1}{18}+\frac{1}{3}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Least common multiple of 18 and 3 is 18. Convert \frac{1}{18} and \frac{1}{3} to fractions with denominator 18.

$$sort(20,\sqrt{\left(\frac{3}{16}+\left(\frac{1}{18}+\frac{6}{18}\right)\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Since \frac{1}{18} and \frac{6}{18} have the same denominator, add them by adding their numerators.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{1+6}{18}\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Add 1 and 6 to get 7.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{7}{18}\times \frac{9}{7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Multiply \frac{7}{18} times \frac{9}{7} by multiplying numerator times numerator and denominator times denominator.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{7\times 9}{18\times 7}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Cancel out 7 in both numerator and denominator.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{9}{18}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Reduce the fraction \frac{9}{18} to lowest terms by extracting and canceling out 9.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{1}{2}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Least common multiple of 16 and 2 is 16. Convert \frac{3}{16} and \frac{1}{2} to fractions with denominator 16.

$$sort(20,\sqrt{\left(\frac{3}{16}+\frac{8}{16}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Since \frac{3}{16} and \frac{8}{16} have the same denominator, add them by adding their numerators.

$$sort(20,\sqrt{\left(\frac{3+8}{16}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Add 3 and 8 to get 11.

$$sort(20,\sqrt{\left(\frac{11}{16}+\frac{1}{2^{2}}\right)\times \frac{1}{15}})$$

Calculate 2 to the power of 2 and get 4.

$$sort(20,\sqrt{\left(\frac{11}{16}+\frac{1}{4}\right)\times \frac{1}{15}})$$

Least common multiple of 16 and 4 is 16. Convert \frac{11}{16} and \frac{1}{4} to fractions with denominator 16.

$$sort(20,\sqrt{\left(\frac{11}{16}+\frac{4}{16}\right)\times \frac{1}{15}})$$

Since \frac{11}{16} and \frac{4}{16} have the same denominator, add them by adding their numerators.

$$sort(20,\sqrt{\frac{11+4}{16}\times \frac{1}{15}})$$

Add 11 and 4 to get 15.

$$sort(20,\sqrt{\frac{15}{16}\times \frac{1}{15}})$$

Multiply \frac{15}{16} times \frac{1}{15} by multiplying numerator times numerator and denominator times denominator.

$$sort(20,\sqrt{\frac{15\times 1}{16\times 15}})$$

Cancel out 15 in both numerator and denominator.

$$sort(20,\sqrt{\frac{1}{16}})$$

Rewrite the square root of the division \frac{1}{16} as the division of square roots \frac{\sqrt{1}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$sort(20,\frac{1}{4})$$

Convert decimal numbers in the list 20,\frac{1}{4} to fractions.

$$20,\frac{1}{4}$$

To sort the list, start from a single element 20.

$$20$$

Insert \frac{1}{4} to the appropriate location in the new list.

$$\frac{1}{4},20$$

Evaluate

$$20,\ \frac{1}{4}$$