2. Simplify the following a) \( \frac{32 \mathrm{atc}}{54 \mathrm{ac}} \times \frac{108}{45^{2}} \) b) \( \frac{24 x^{2} y}{2 x y} \times \frac{24 x y^{2}}{-2 z x} \) c) \( \frac{12 x}{y} \div \frac{6 x}{y} \) d) \( \frac{72 \pi+7}{-5 n} \div \frac{18-r}{36 n^{2}} \)

a) We start with \[ \frac{32\,\mathrm{atc}}{54\,\mathrm{ac}} \times \frac{108}{45^2}. \] First, simplify the first fraction: \[ \frac{32\,\mathrm{atc}}{54\,\mathrm{ac}} = \frac{32}{54}\cdot \frac{\mathrm{atc}}{\mathrm{ac}}. \] Cancel the common factors \(\mathrm{a}\) and \(\mathrm{c}\) (assuming these are nonzero), leaving \[ \frac{32}{54}\,t. \] Divide numerator and denominator by \(2\): \[ \frac{16}{27}\,t. \] Next, note that \[ 45^2 = 2025, \] so the second fraction is \[ \frac{108}{2025}. \] Thus, we have \[ \frac{16\,t}{27} \times \frac{108}{2025}. \] Simplify by canceling the factor \(27\) into \(108\) since \(108\div27=4\): \[ \frac{16\,t \times 4}{2025} = \frac{64\,t}{2025}. \] --- b) We simplify \[ \frac{24 x^{2} y}{2 x y} \times \frac{24 x y^{2}}{-2 z x}. \] First, simplify the first fraction: \[ \frac{24 x^{2} y}{2 x y} = \frac{24}{2}\cdot \frac{x^{2}}{x}\cdot \frac{y}{y} = 12x. \] For the second fraction: \[ \frac{24 x y^{2}}{-2 z x} = \frac{24}{-2}\cdot \frac{x}{x}\cdot \frac{y^{2}}{z} = -12\,\frac{y^{2}}{z}. \] Multiplying the simplified expressions: \[ 12x \times \left(-\frac{12\,y^{2}}{z}\right) = -144\,\frac{x y^{2}}{z}. \] --- c) We have \[ \frac{12 x}{y} \div \frac{6 x}{y}. \] Division by a fraction is equivalent to multiplication by its reciprocal: \[ \frac{12 x}{y} \times \frac{y}{6 x} = \frac{12x\,\cancel{y}}{6x\,\cancel{y}} = \frac{12}{6} = 2. \] --- d) We simplify \[ \frac{72 \pi+7}{-5 n} \div \frac{18-r}{36 n^{2}}. \] Convert the division to multiplication by the reciprocal: \[ \frac{72 \pi+7}{-5 n} \times \frac{36 n^{2}}{18-r}. \] Combine the expressions: \[ \frac{(72 \pi+7)\cdot 36 n^{2}}{-5 n (18-r)}. \] Cancel a factor \(n\) (with \(n\neq 0\)): \[ \frac{(72 \pi+7)\cdot 36 n}{-5 (18-r)}. \] Recognize that a negative in the denominator can be rewritten so that \[ -5 (18-r) = 5(r-18), \] thus, \[ \frac{(72 \pi+7)\cdot 36 n}{-5 (18-r)} = \frac{36 n (72\pi+7)}{5(r-18)}. \] --- Final Answers: a) \(\displaystyle \frac{64\,t}{2025}\) b) \(\displaystyle -\frac{144 x y^{2}}{z}\) c) \(2\) d) \(\displaystyle \frac{36 n (72\pi+7)}{5(r-18)}\)