Solve {c}{[(1+2/3)*1/5+(4/3)^{2}*15/6-(frac{1}{3})^{2}:frac{1}{3}]:frac{5}{2}}{frac{2}{11^{2}-9^2-3*5}+frac{5}{sqrt{13^2-12^2-3^2}}-1}

$$\left. \begin{array} { c } { [ ( 1 + \frac { 2 } { 3 } ) \times \frac { 1 } { 5 } + ( \frac { 4 } { 3 } ) ^ { 2 } \times \frac { 15 } { 6 } - ( \frac { 1 } { 3 } ) ^ { 2 } : \frac { 1 } { 3 } ] : \frac { 5 } { 2 } } \\ { \frac { 2 } { 11 ^ { 2 } - 9 ^ { 2 } - 3 \times 5 } + \frac { 5 } { \sqrt { 13 ^ { 2 } - 12 ^ { 2 } - 3 ^ { 2 } } } - 1 } \end{array} \right.$$

Sort

$$\frac{33}{100},\frac{16}{9}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 2 to get 1.

$$sort(\frac{\left(1+\frac{2}{3}\right)\times \frac{1}{5}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Convert 1 to fraction \frac{3}{3}.

$$sort(\frac{\left(\frac{3}{3}+\frac{2}{3}\right)\times \frac{1}{5}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

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

$$sort(\frac{\frac{3+2}{3}\times \frac{1}{5}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Add 3 and 2 to get 5.

$$sort(\frac{\frac{5}{3}\times \frac{1}{5}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Multiply \frac{5}{3} times \frac{1}{5} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{\frac{5\times 1}{3\times 5}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Cancel out 5 in both numerator and denominator.

$$sort(\frac{\frac{1}{3}+\left(\frac{4}{3}\right)^{2}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Calculate \frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$sort(\frac{\frac{1}{3}+\frac{16}{9}\times \frac{15}{6}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Reduce the fraction \frac{15}{6} to lowest terms by extracting and canceling out 3.

$$sort(\frac{\frac{1}{3}+\frac{16}{9}\times \frac{5}{2}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Multiply \frac{16}{9} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{\frac{1}{3}+\frac{16\times 5}{9\times 2}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Do the multiplications in the fraction \frac{16\times 5}{9\times 2}.

$$sort(\frac{\frac{1}{3}+\frac{80}{18}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Reduce the fraction \frac{80}{18} to lowest terms by extracting and canceling out 2.

$$sort(\frac{\frac{1}{3}+\frac{40}{9}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Least common multiple of 3 and 9 is 9. Convert \frac{1}{3} and \frac{40}{9} to fractions with denominator 9.

$$sort(\frac{\frac{3}{9}+\frac{40}{9}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Since \frac{3}{9} and \frac{40}{9} have the same denominator, add them by adding their numerators.

$$sort(\frac{\frac{3+40}{9}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Add 3 and 40 to get 43.

$$sort(\frac{\frac{43}{9}-\left(\frac{1}{3}\right)^{1}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Calculate \frac{1}{3} to the power of 1 and get \frac{1}{3}.

$$sort(\frac{\frac{43}{9}-\frac{1}{3}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Least common multiple of 9 and 3 is 9. Convert \frac{43}{9} and \frac{1}{3} to fractions with denominator 9.

$$sort(\frac{\frac{43}{9}-\frac{3}{9}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Since \frac{43}{9} and \frac{3}{9} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{\frac{43-3}{9}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Subtract 3 from 43 to get 40.

$$sort(\frac{\frac{40}{9}}{\frac{5}{2}},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Divide \frac{40}{9} by \frac{5}{2} by multiplying \frac{40}{9} by the reciprocal of \frac{5}{2}.

$$sort(\frac{40}{9}\times \frac{2}{5},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Multiply \frac{40}{9} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{40\times 2}{9\times 5},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Do the multiplications in the fraction \frac{40\times 2}{9\times 5}.

$$sort(\frac{80}{45},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Reduce the fraction \frac{80}{45} to lowest terms by extracting and canceling out 5.

$$sort(\frac{16}{9},\frac{2}{11^{2}-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Calculate 11 to the power of 2 and get 121.

$$sort(\frac{16}{9},\frac{2}{121-9^{2}-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Calculate 9 to the power of 2 and get 81.

$$sort(\frac{16}{9},\frac{2}{121-81-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Subtract 81 from 121 to get 40.

$$sort(\frac{16}{9},\frac{2}{40-3\times 5}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Multiply 3 and 5 to get 15.

$$sort(\frac{16}{9},\frac{2}{40-15}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Subtract 15 from 40 to get 25.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{13^{2}-12^{2}-3^{2}}}-1)$$

Calculate 13 to the power of 2 and get 169.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{169-12^{2}-3^{2}}}-1)$$

Calculate 12 to the power of 2 and get 144.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{169-144-3^{2}}}-1)$$

Subtract 144 from 169 to get 25.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{25-3^{2}}}-1)$$

Calculate 3 to the power of 2 and get 9.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{25-9}}-1)$$

Subtract 9 from 25 to get 16.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{\sqrt{16}}-1)$$

Calculate the square root of 16 and get 4.

$$sort(\frac{16}{9},\frac{2}{25}+\frac{5}{4}-1)$$

Least common multiple of 25 and 4 is 100. Convert \frac{2}{25} and \frac{5}{4} to fractions with denominator 100.

$$sort(\frac{16}{9},\frac{8}{100}+\frac{125}{100}-1)$$

Since \frac{8}{100} and \frac{125}{100} have the same denominator, add them by adding their numerators.

$$sort(\frac{16}{9},\frac{8+125}{100}-1)$$

Add 8 and 125 to get 133.

$$sort(\frac{16}{9},\frac{133}{100}-1)$$

Convert 1 to fraction \frac{100}{100}.

$$sort(\frac{16}{9},\frac{133}{100}-\frac{100}{100})$$

Since \frac{133}{100} and \frac{100}{100} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{16}{9},\frac{133-100}{100})$$

Subtract 100 from 133 to get 33.

$$sort(\frac{16}{9},\frac{33}{100})$$

Least common denominator of the numbers in the list \frac{16}{9},\frac{33}{100} is 900. Convert numbers in the list to fractions with denominator 900.

$$\frac{1600}{900},\frac{297}{900}$$

To sort the list, start from a single element \frac{1600}{900}.

$$\frac{1600}{900}$$

Insert \frac{297}{900} to the appropriate location in the new list.

$$\frac{297}{900},\frac{1600}{900}$$

Replace the obtained fractions with the initial values.

$$\frac{33}{100},\frac{16}{9}$$

Evaluate

$$\frac{16}{9},\ \frac{33}{100}$$