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  miniaufgabe.js
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==== Montag 15. Dezember 2025 ====
Ausrechnen, Resultat als vollständig gekürzter Bruch. **Achtung:** Potenzen erst ganz am Schluss ausrechnen. Zuerst Basen in Primfaktoren zerlegen und vor dem Multiplizieren kürzen!<JS>miniAufgabe("#exonegativeexponenten","#solnegativeexponenten",
[["$\\displaystyle \\left(\\frac{4}{3}-\\frac{14}{15}\\right)^{-2} \\cdot \\left(-\\frac{5}{4}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{4}{3}-\\frac{14}{15}\\right)^{-2} \\cdot \\left(-\\frac{5}{4}\\right)^{-3} = \\left(\\frac{20}{15}-\\frac{14}{15}\\right)^{-2} \\cdot \\left(-\\frac{5}{4}\\right)^{-3} = \\left(\\frac{2}{5}\\right)^{-2} \\cdot \\left(-\\frac{4}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{4}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{2^{2}}{5}\\right)^3 = \\frac{5^{2}}{2^{2}} \\cdot -\\frac{2^{6}}{5^{3}} = -\\frac{2^{4}}{5} = -\\frac{16}{5}$"], ["$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(\\frac{5}{6}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(\\frac{5}{6}\\right)^{-3} = \\left(\\frac{5}{10}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(\\frac{5}{6}\\right)^{-3} = \\left(\\frac{2}{5}\\right)^{-2} \\cdot \\left(\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(\\frac{2 \\cdot 3}{5}\\right)^3 = \\frac{5^{2}}{2^{2}} \\cdot \\frac{2^{3} \\cdot 3^{3}}{5^{3}} = \\frac{2 \\cdot 3^{3}}{5} = \\frac{54}{5}$"], ["$\\displaystyle \\left(-\\frac{6}{7}+\\frac{83}{56}\\right)^{-2} \\cdot \\left(\\frac{6}{5}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{6}{7}+\\frac{83}{56}\\right)^{-2} \\cdot \\left(\\frac{6}{5}\\right)^{-3} = \\left(-\\frac{48}{56}+\\frac{83}{56}\\right)^{-2} \\cdot \\left(\\frac{6}{5}\\right)^{-3} = \\left(\\frac{5}{8}\\right)^{-2} \\cdot \\left(\\frac{5}{6}\\right)^3 = \\left(\\frac{8}{5}\\right)^2 \\cdot \\left(\\frac{5}{6}\\right)^3 = \\left(\\frac{2^{3}}{5}\\right)^2 \\cdot \\left(\\frac{5}{2 \\cdot 3}\\right)^3 = \\frac{2^{6}}{5^{2}} \\cdot \\frac{5^{3}}{2^{3} \\cdot 3^{3}} = \\frac{2^{3} \\cdot 5}{3^{3}} = \\frac{40}{27}$"], ["$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3} = \\left(\\frac{5}{10}-\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3} = \\left(\\frac{2}{5}\\right)^{-2} \\cdot \\left(-\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{2 \\cdot 3}{5}\\right)^3 = \\frac{5^{2}}{2^{2}} \\cdot -\\frac{2^{3} \\cdot 3^{3}}{5^{3}} = -\\frac{2 \\cdot 3^{3}}{5} = -\\frac{54}{5}$"], ["$\\displaystyle \\left(\\frac{5}{3}-\\frac{34}{15}\\right)^{-2} \\cdot \\left(-\\frac{10}{9}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{5}{3}-\\frac{34}{15}\\right)^{-2} \\cdot \\left(-\\frac{10}{9}\\right)^{-3} = \\left(\\frac{25}{15}-\\frac{34}{15}\\right)^{-2} \\cdot \\left(-\\frac{10}{9}\\right)^{-3} = \\left(-\\frac{3}{5}\\right)^{-2} \\cdot \\left(-\\frac{9}{10}\\right)^3 = \\left(\\frac{5}{3}\\right)^2 \\cdot \\left(-\\frac{9}{10}\\right)^3 = \\left(\\frac{5}{3}\\right)^2 \\cdot \\left(-\\frac{3^{2}}{2 \\cdot 5}\\right)^3 = \\frac{5^{2}}{3^{2}} \\cdot -\\frac{3^{6}}{2^{3} \\cdot 5^{3}} = -\\frac{3^{4}}{2^{3} \\cdot 5} = -\\frac{81}{40}$"], ["$\\displaystyle \\left(-\\frac{6}{5}+\\frac{3}{2}\\right)^{-2} \\cdot \\left(-\\frac{5}{2}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{6}{5}+\\frac{3}{2}\\right)^{-2} \\cdot \\left(-\\frac{5}{2}\\right)^{-3} = \\left(-\\frac{12}{10}+\\frac{15}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{2}\\right)^{-3} = \\left(\\frac{3}{10}\\right)^{-2} \\cdot \\left(-\\frac{2}{5}\\right)^3 = \\left(\\frac{10}{3}\\right)^2 \\cdot \\left(-\\frac{2}{5}\\right)^3 = \\left(\\frac{2 \\cdot 5}{3}\\right)^2 \\cdot \\left(-\\frac{2}{5}\\right)^3 = \\frac{2^{2} \\cdot 5^{2}}{3^{2}} \\cdot -\\frac{2^{3}}{5^{3}} = -\\frac{2^{5}}{3^{2} \\cdot 5} = -\\frac{32}{45}$"], ["$\\displaystyle \\left(-\\frac{3}{2}+\\frac{2}{3}\\right)^{-2} \\cdot \\left(-\\frac{3}{5}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{3}{2}+\\frac{2}{3}\\right)^{-2} \\cdot \\left(-\\frac{3}{5}\\right)^{-3} = \\left(-\\frac{9}{6}+\\frac{4}{6}\\right)^{-2} \\cdot \\left(-\\frac{3}{5}\\right)^{-3} = \\left(-\\frac{5}{6}\\right)^{-2} \\cdot \\left(-\\frac{5}{3}\\right)^3 = \\left(\\frac{6}{5}\\right)^2 \\cdot \\left(-\\frac{5}{3}\\right)^3 = \\left(\\frac{2 \\cdot 3}{5}\\right)^2 \\cdot \\left(-\\frac{5}{3}\\right)^3 = \\frac{2^{2} \\cdot 3^{2}}{5^{2}} \\cdot -\\frac{5^{3}}{3^{3}} = -\\frac{2^{2} \\cdot 5}{3} = -\\frac{20}{3}$"], ["$\\displaystyle \\left(-\\frac{2}{3}+\\frac{3}{2}\\right)^{-2} \\cdot \\left(-\\frac{9}{5}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{2}{3}+\\frac{3}{2}\\right)^{-2} \\cdot \\left(-\\frac{9}{5}\\right)^{-3} = \\left(-\\frac{4}{6}+\\frac{9}{6}\\right)^{-2} \\cdot \\left(-\\frac{9}{5}\\right)^{-3} = \\left(\\frac{5}{6}\\right)^{-2} \\cdot \\left(-\\frac{5}{9}\\right)^3 = \\left(\\frac{6}{5}\\right)^2 \\cdot \\left(-\\frac{5}{9}\\right)^3 = \\left(\\frac{2 \\cdot 3}{5}\\right)^2 \\cdot \\left(-\\frac{5}{3^{2}}\\right)^3 = \\frac{2^{2} \\cdot 3^{2}}{5^{2}} \\cdot -\\frac{5^{3}}{3^{6}} = -\\frac{2^{2} \\cdot 5}{3^{4}} = -\\frac{20}{81}$"], ["$\\displaystyle \\left(-\\frac{2}{5}+\\frac{4}{35}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{2}{5}+\\frac{4}{35}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3} = \\left(-\\frac{14}{35}+\\frac{4}{35}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3} = \\left(-\\frac{2}{7}\\right)^{-2} \\cdot \\left(-\\frac{6}{7}\\right)^3 = \\left(\\frac{7}{2}\\right)^2 \\cdot \\left(-\\frac{6}{7}\\right)^3 = \\left(\\frac{7}{2}\\right)^2 \\cdot \\left(-\\frac{2 \\cdot 3}{7}\\right)^3 = \\frac{7^{2}}{2^{2}} \\cdot -\\frac{2^{3} \\cdot 3^{3}}{7^{3}} = -\\frac{2 \\cdot 3^{3}}{7} = -\\frac{54}{7}$"], ["$\\displaystyle \\left(\\frac{3}{4}-\\frac{17}{12}\\right)^{-2} \\cdot \\left(-\\frac{3}{4}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{3}{4}-\\frac{17}{12}\\right)^{-2} \\cdot \\left(-\\frac{3}{4}\\right)^{-3} = \\left(\\frac{9}{12}-\\frac{17}{12}\\right)^{-2} \\cdot \\left(-\\frac{3}{4}\\right)^{-3} = \\left(-\\frac{2}{3}\\right)^{-2} \\cdot \\left(-\\frac{4}{3}\\right)^3 = \\left(\\frac{3}{2}\\right)^2 \\cdot \\left(-\\frac{4}{3}\\right)^3 = \\left(\\frac{3}{2}\\right)^2 \\cdot \\left(-\\frac{2^{2}}{3}\\right)^3 = \\frac{3^{2}}{2^{2}} \\cdot -\\frac{2^{6}}{3^{3}} = -\\frac{2^{4}}{3} = -\\frac{16}{3}$"]],
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