4 1. \( \frac{\left(3 a^{9} b^{2} c\right)^{2}}{6 a^{4} c^{5}} \div \frac{2 a^{2} b r^{4}}{20 a b c} \)

To simplify the expression step by step: 1. First, simplify the numerator and denominator in the first fraction: \(\left(3 a^{9} b^{2} c\right)^{2} = 9 a^{18} b^{4} c^{2}\). So, the first fraction becomes: \(\frac{9 a^{18} b^{4} c^{2}}{6 a^{4} c^{5}}\). We can simplify this: \(\frac{9}{6} = \frac{3}{2}\), and for the variable terms: \(a^{18}/a^{4} = a^{14}\), \(c^{2}/c^{5} = c^{-3} = \frac{1}{c^{3}}\). Thus, it simplifies to: \(\frac{3 a^{14} b^{4}}{2 c^{3}}\). 2. Now simplify the second fraction: \(\frac{2 a^{2} b r^{4}}{20 a b c} = \frac{2}{20} \cdot \frac{a^{2}}{a} \cdot \frac{b}{b} \cdot \frac{r^{4}}{c} = \frac{1}{10} a^{1} r^{4} c^{-1} = \frac{1 a r^{4}}{10 c}\). 3. Now, divide the simplified first fraction by the second: \(\frac{3 a^{14} b^{4}}{2 c^{3}} \div \frac{a r^{4}}{10 c} = \frac{3 a^{14} b^{4}}{2 c^{3}} \times \frac{10 c}{a r^{4}} = \frac{30 a^{13} b^{4}}{2 r^{4} c^{2}} = \frac{15 a^{13} b^{4}}{r^{4} c^{2}}\). So, the final result is: \(\frac{15 a^{13} b^{4}}{r^{4} c^{2}}\).