1. **Énoncé du problème :** Calculer les produits de fractions donnés et exprimer les résultats sous forme de fractions simplifiées. 2. **Rappel de la règle de multiplication des fractions :** Pour multiplier deux fractions $\frac{a}{b}$ et $\frac{c}{d}$, on multiplie les numérateurs entre eux et les dénominateurs entre eux : $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Calculs détaillés :** - $A = \frac{2}{7} \times \frac{4}{3} = \frac{2 \times 4}{7 \times 3} = \frac{8}{21}$ - $B = \frac{4}{7} \times \frac{2}{3} = \frac{4 \times 2}{7 \times 3} = \frac{8}{21}$ - $C = 7 \times \frac{4}{11} = \frac{7}{1} \times \frac{4}{11} = \frac{28}{11}$ - $D = \frac{7}{11} \times 4 = \frac{7}{11} \times \frac{4}{1} = \frac{28}{11}$ - $E = 9 \times \left(-\frac{4}{5}\right) = \frac{9}{1} \times \left(-\frac{4}{5}\right) = -\frac{36}{5}$ - $F = -\frac{2}{5} \times \frac{9}{5} = \frac{-2 \times 9}{5 \times 5} = -\frac{18}{25}$ - $G = -\frac{7}{6} \times \frac{5}{-9} = \frac{-7 \times 5}{6 \times -9} = \frac{-35}{-54} = \frac{35}{54}$ - $H = \frac{7}{-10} \times \frac{-11}{-3} = \frac{7 \times -11}{-10 \times -3} = \frac{-77}{30}$ - $I = \frac{-11}{-4} \times \frac{-9}{-13} = \frac{11}{4} \times \frac{9}{13} = \frac{99}{52}$ - $J = \frac{-5}{-7} \times \left(\frac{15}{-2}\right) = \frac{5}{7} \times \left(-\frac{15}{2}\right) = -\frac{75}{14}$ - $K = -\frac{5}{2} \times \frac{2}{-3} = \frac{-5 \times 2}{2 \times -3} = \frac{-10}{-6} = \frac{5}{3}$ - $L = \frac{-2}{-3} \times \left(\frac{-3}{-7}\right) = \frac{2}{3} \times \frac{3}{7} = \frac{6}{21} = \frac{2}{7}$ - $M = 4 \times \frac{5}{-4} = \frac{4}{1} \times \left(-\frac{5}{4}\right) = -\frac{20}{4} = -5$ - $N = -\frac{4}{15} \times (-5) = -\frac{4}{15} \times \left(-\frac{5}{1}\right) = \frac{20}{15} = \frac{4}{3}$ - $O = -12 \times \left(-\frac{7}{-6}\right) = -12 \times \left(-\frac{7}{-6}\right) = -12 \times \frac{7}{6} = -14$ - $P = \frac{-2}{-3} \times \frac{5}{-4} = \frac{2}{3} \times \left(-\frac{5}{4}\right) = -\frac{10}{12} = -\frac{5}{6}$ - $Q = \frac{5}{-7} \times \frac{-3}{-15} = \left(-\frac{5}{7}\right) \times \left(-\frac{3}{15}\right) = \frac{15}{105} = \frac{1}{7}$ - $R = \frac{-5}{-7} \times \frac{14}{-15} = \frac{5}{7} \times \left(-\frac{14}{15}\right) = -\frac{70}{105} = -\frac{2}{3}$ - $S = \frac{6}{-10} \times \frac{-1}{-3} = \left(-\frac{3}{5}\right) \times \left(-\frac{1}{3}\right) = \frac{3}{15} = \frac{1}{5}$ - $T = \frac{-28}{-21} \times \left(\frac{-6}{-4}\right) = \frac{28}{21} \times \frac{6}{4} = \frac{168}{84} = 2$ 4. **Réponses finales :** $A=\frac{8}{21}, B=\frac{8}{21}, C=\frac{28}{11}, D=\frac{28}{11}, E=-\frac{36}{5}, F=-\frac{18}{25}, G=\frac{35}{54}, H=-\frac{77}{30}, I=\frac{99}{52}, J=-\frac{75}{14}, K=\frac{5}{3}, L=\frac{2}{7}, M=-5, N=\frac{4}{3}, O=-14, P=-\frac{5}{6}, Q=\frac{1}{7}, R=-\frac{2}{3}, S=\frac{1}{5}, T=2$