Solve {l}{1/2:1/6/(2*frac{1{6})^{2}}(2*3/4)^{-2}(frac{4}{2})^-2}{(2*frac{1}{6})^2}{frac{1}{6}:frac{1}{6}(-frac{1}{6}*4)^2:(-frac{2}{3})^2}

$$\left. \begin{array} { l } { \frac { \frac { 1 } { 2 } : \frac { 1 } { 6 } } { ( 2 \cdot \frac { 1 } { 6 } ) ^ { 2 } } ( 2 \cdot \frac { 3 } { 4 } ) ^ { - 2 } ( \frac { 4 } { 2 } ) ^ { - 2 } } \\ { ( 2 \cdot \frac { 1 } { 6 } ) ^ { 2 } } \\ { \frac { 1 } { 6 } : \frac { 1 } { 6 } ( - \frac { 1 } { 6 } \cdot 4 ) ^ { 2 } : ( - \frac { 2 } { 3 } ) ^ { 2 } } \end{array} \right.$$

Sort

$$\frac{1}{9},1,3$$

Solution Steps

Divide \frac{1}{6} by \frac{1}{6} to get 1.

$$sort(\frac{\frac{\frac{1}{2}}{\frac{1}{6}}}{\left(2\times \frac{1}{6}\right)^{2}}\times \left(2\times \frac{3}{4}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Divide \frac{1}{2} by \frac{1}{6} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{6}.

$$sort(\frac{\frac{1}{2}\times 6}{\left(2\times \frac{1}{6}\right)^{2}}\times \left(2\times \frac{3}{4}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply \frac{1}{2} and 6 to get 3.

$$sort(\frac{3}{\left(2\times \frac{1}{6}\right)^{2}}\times \left(2\times \frac{3}{4}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply 2 and \frac{1}{6} to get \frac{1}{3}.

$$sort(\frac{3}{\left(\frac{1}{3}\right)^{2}}\times \left(2\times \frac{3}{4}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

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

Divide 3 by \frac{1}{9} by multiplying 3 by the reciprocal of \frac{1}{9}.

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

Multiply 3 and 9 to get 27.

$$sort(27\times \left(2\times \frac{3}{4}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply 2 and \frac{3}{4} to get \frac{3}{2}.

$$sort(27\times \left(\frac{3}{2}\right)^{-2}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

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

$$sort(27\times \frac{4}{9}\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply 27 and \frac{4}{9} to get 12.

$$sort(12\times \left(\frac{4}{2}\right)^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Divide 4 by 2 to get 2.

$$sort(12\times 2^{-2},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$sort(12\times \frac{1}{4},\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply 12 and \frac{1}{4} to get 3.

$$sort(3,\left(2\times \frac{1}{6}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Multiply 2 and \frac{1}{6} to get \frac{1}{3}.

$$sort(3,\left(\frac{1}{3}\right)^{2},\frac{1\left(-\frac{1}{6}\times 4\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

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

Multiply -\frac{1}{6} and 4 to get -\frac{2}{3}.

$$sort(3,\frac{1}{9},\frac{1\left(-\frac{2}{3}\right)^{2}}{\left(-\frac{2}{3}\right)^{2}})$$

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

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

Multiply 1 and \frac{4}{9} to get \frac{4}{9}.

$$sort(3,\frac{1}{9},\frac{\frac{4}{9}}{\left(-\frac{2}{3}\right)^{2}})$$

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

$$sort(3,\frac{1}{9},\frac{\frac{4}{9}}{\frac{4}{9}})$$

Divide \frac{4}{9} by \frac{4}{9} to get 1.

$$sort(3,\frac{1}{9},1)$$

Convert decimal numbers in the list 3,\frac{1}{9},1 to fractions.

$$3,\frac{1}{9},1$$

To sort the list, start from a single element 3.

$$3$$

Insert \frac{1}{9} to the appropriate location in the new list.

$$\frac{1}{9},3$$

Insert 1 to the appropriate location in the new list.

$$\frac{1}{9},1,3$$

Evaluate

$$3,\ \frac{1}{9},\ 1$$