miniaufgabe.js
==== 25. Oktober 2021 bis 29. Oktober 2021 ====
=== Donnerstag 28. Oktober 2021 ===
Ausrechnen, Resultat als gekürzter Bruch:miniAufgabe("#exokettenbruch","#solkettenbruch",
[["$\\displaystyle 2 + \\frac{1}{3 + \\frac{1}{2 + \\frac{1}{4}}}$", "$\\displaystyle 2 + \\frac{1}{3 + \\frac{1}{2 + \\frac{1}{4}}} = 2 + \\frac{1}{3 + \\frac{1}{\\frac{9}{4}}} = 2 + \\frac{1}{3 + \\frac{4}{9}} = 2 + \\frac{1}{\\frac{31}{9}} = 2 + \\frac{9}{31} = \\frac{71}{31}$"], ["$\\displaystyle 2 + \\frac{1}{3 + \\frac{1}{3 + \\frac{1}{3}}}$", "$\\displaystyle 2 + \\frac{1}{3 + \\frac{1}{3 + \\frac{1}{3}}} = 2 + \\frac{1}{3 + \\frac{1}{\\frac{10}{3}}} = 2 + \\frac{1}{3 + \\frac{3}{10}} = 2 + \\frac{1}{\\frac{33}{10}} = 2 + \\frac{10}{33} = \\frac{76}{33}$"], ["$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{2 + \\frac{1}{3}}}$", "$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{2 + \\frac{1}{3}}} = 1 + \\frac{1}{3 + \\frac{1}{\\frac{7}{3}}} = 1 + \\frac{1}{3 + \\frac{3}{7}} = 1 + \\frac{1}{\\frac{24}{7}} = 1 + \\frac{7}{24} = \\frac{31}{24}$"], ["$\\displaystyle 2 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{4}}}$", "$\\displaystyle 2 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{4}}} = 2 + \\frac{1}{2 + \\frac{1}{\\frac{17}{4}}} = 2 + \\frac{1}{2 + \\frac{4}{17}} = 2 + \\frac{1}{\\frac{38}{17}} = 2 + \\frac{17}{38} = \\frac{93}{38}$"], ["$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{3 + \\frac{1}{4}}}$", "$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{3 + \\frac{1}{4}}} = 1 + \\frac{1}{3 + \\frac{1}{\\frac{13}{4}}} = 1 + \\frac{1}{3 + \\frac{4}{13}} = 1 + \\frac{1}{\\frac{43}{13}} = 1 + \\frac{13}{43} = \\frac{56}{43}$"], ["$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{1 + \\frac{1}{5}}}$", "$\\displaystyle 1 + \\frac{1}{3 + \\frac{1}{1 + \\frac{1}{5}}} = 1 + \\frac{1}{3 + \\frac{1}{\\frac{6}{5}}} = 1 + \\frac{1}{3 + \\frac{5}{6}} = 1 + \\frac{1}{\\frac{23}{6}} = 1 + \\frac{6}{23} = \\frac{29}{23}$"], ["$\\displaystyle 1 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{3}}}$", "$\\displaystyle 1 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{3}}} = 1 + \\frac{1}{2 + \\frac{1}{\\frac{13}{3}}} = 1 + \\frac{1}{2 + \\frac{3}{13}} = 1 + \\frac{1}{\\frac{29}{13}} = 1 + \\frac{13}{29} = \\frac{42}{29}$"], ["$\\displaystyle 2 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{3}}}$", "$\\displaystyle 2 + \\frac{1}{2 + \\frac{1}{4 + \\frac{1}{3}}} = 2 + \\frac{1}{2 + \\frac{1}{\\frac{13}{3}}} = 2 + \\frac{1}{2 + \\frac{3}{13}} = 2 + \\frac{1}{\\frac{29}{13}} = 2 + \\frac{13}{29} = \\frac{71}{29}$"], ["$\\displaystyle 1 + \\frac{1}{1 + \\frac{1}{3 + \\frac{1}{5}}}$", "$\\displaystyle 1 + \\frac{1}{1 + \\frac{1}{3 + \\frac{1}{5}}} = 1 + \\frac{1}{1 + \\frac{1}{\\frac{16}{5}}} = 1 + \\frac{1}{1 + \\frac{5}{16}} = 1 + \\frac{1}{\\frac{21}{16}} = 1 + \\frac{16}{21} = \\frac{37}{21}$"], ["$\\displaystyle 1 + \\frac{1}{1 + \\frac{1}{4 + \\frac{1}{5}}}$", "$\\displaystyle 1 + \\frac{1}{1 + \\frac{1}{4 + \\frac{1}{5}}} = 1 + \\frac{1}{1 + \\frac{1}{\\frac{21}{5}}} = 1 + \\frac{1}{1 + \\frac{5}{21}} = 1 + \\frac{1}{\\frac{26}{21}} = 1 + \\frac{21}{26} = \\frac{47}{26}$"]],
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=== Freitag 29. Oktober 2021 ===
Ausrechnen, Resultat als vollständig gekürzter Bruch. **Achtung:** Potenzen erst ganz am Schluss ausrechnen. Zuerst Basen in Primfaktoren zerlegen und vor dem Multiplizieren kürzen!miniAufgabe("#exonegativeexponenten","#solnegativeexponenten",
[["$\\displaystyle \\left(\\frac{3}{2}-\\frac{11}{10}\\right)^{-2} \\cdot \\left(-\\frac{3}{2}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{3}{2}-\\frac{11}{10}\\right)^{-2} \\cdot \\left(-\\frac{3}{2}\\right)^{-3} = \\left(\\frac{15}{10}-\\frac{11}{10}\\right)^{-2} \\cdot \\left(-\\frac{3}{2}\\right)^{-3} = \\left(\\frac{2}{5}\\right)^{-2} \\cdot \\left(-\\frac{2}{3}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{2}{3}\\right)^3 = \\left(\\frac{5}{2}\\right)^2 \\cdot \\left(-\\frac{2}{3}\\right)^3 = \\frac{5^{2}}{2^{2}} \\cdot -\\frac{2^{3}}{3^{3}} = -\\frac{2 \\cdot 5^{2}}{3^{3}} = -\\frac{50}{27}$"], ["$\\displaystyle \\left(\\frac{1}{2}-\\frac{4}{3}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{2}-\\frac{4}{3}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3} = \\left(\\frac{3}{6}-\\frac{8}{6}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3} = \\left(-\\frac{5}{6}\\right)^{-2} \\cdot \\left(\\frac{5}{4}\\right)^3 = \\left(\\frac{6}{5}\\right)^2 \\cdot \\left(\\frac{5}{4}\\right)^3 = \\left(\\frac{2 \\cdot 3}{5}\\right)^2 \\cdot \\left(\\frac{5}{2^{2}}\\right)^3 = \\frac{2^{2} \\cdot 3^{2}}{5^{2}} \\cdot \\frac{5^{3}}{2^{6}} = \\frac{3^{2} \\cdot 5}{2^{4}} = \\frac{45}{16}$"], ["$\\displaystyle \\left(-\\frac{5}{6}+\\frac{4}{3}\\right)^{-2} \\cdot \\left(-\\frac{4}{3}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{5}{6}+\\frac{4}{3}\\right)^{-2} \\cdot \\left(-\\frac{4}{3}\\right)^{-3} = \\left(-\\frac{5}{6}+\\frac{8}{6}\\right)^{-2} \\cdot \\left(-\\frac{4}{3}\\right)^{-3} = \\left(\\frac{1}{2}\\right)^{-2} \\cdot \\left(-\\frac{3}{4}\\right)^3 = \\left(2\\right)^2 \\cdot \\left(-\\frac{3}{4}\\right)^3 = \\left(\\frac{2}{1}\\right)^2 \\cdot \\left(-\\frac{3}{2^{2}}\\right)^3 = \\frac{2^{2}}{1^{2}} \\cdot -\\frac{3^{3}}{2^{6}} = -\\frac{3^{3}}{1^{2} \\cdot 2^{4}} = -\\frac{27}{16}$"], ["$\\displaystyle \\left(\\frac{5}{6}-\\frac{1}{12}\\right)^{-2} \\cdot \\left(-\\frac{4}{9}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{5}{6}-\\frac{1}{12}\\right)^{-2} \\cdot \\left(-\\frac{4}{9}\\right)^{-3} = \\left(\\frac{10}{12}-\\frac{1}{12}\\right)^{-2} \\cdot \\left(-\\frac{4}{9}\\right)^{-3} = \\left(\\frac{3}{4}\\right)^{-2} \\cdot \\left(-\\frac{9}{4}\\right)^3 = \\left(\\frac{4}{3}\\right)^2 \\cdot \\left(-\\frac{9}{4}\\right)^3 = \\left(\\frac{2^{2}}{3}\\right)^2 \\cdot \\left(-\\frac{3^{2}}{2^{2}}\\right)^3 = \\frac{2^{4}}{3^{2}} \\cdot -\\frac{3^{6}}{2^{6}} = -\\frac{3^{4}}{2^{2}} = -\\frac{81}{4}$"], ["$\\displaystyle \\left(\\frac{2}{9}-\\frac{13}{18}\\right)^{-2} \\cdot \\left(\\frac{4}{3}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{2}{9}-\\frac{13}{18}\\right)^{-2} \\cdot \\left(\\frac{4}{3}\\right)^{-3} = \\left(\\frac{4}{18}-\\frac{13}{18}\\right)^{-2} \\cdot \\left(\\frac{4}{3}\\right)^{-3} = \\left(-\\frac{1}{2}\\right)^{-2} \\cdot \\left(\\frac{3}{4}\\right)^3 = \\left(2\\right)^2 \\cdot \\left(\\frac{3}{4}\\right)^3 = \\left(\\frac{2}{1}\\right)^2 \\cdot \\left(\\frac{3}{2^{2}}\\right)^3 = \\frac{2^{2}}{1^{2}} \\cdot \\frac{3^{3}}{2^{6}} = \\frac{3^{3}}{1^{2} \\cdot 2^{4}} = \\frac{27}{16}$"], ["$\\displaystyle \\left(-\\frac{3}{10}+\\frac{7}{90}\\right)^{-2} \\cdot \\left(-\\frac{9}{4}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{3}{10}+\\frac{7}{90}\\right)^{-2} \\cdot \\left(-\\frac{9}{4}\\right)^{-3} = \\left(-\\frac{27}{90}+\\frac{7}{90}\\right)^{-2} \\cdot \\left(-\\frac{9}{4}\\right)^{-3} = \\left(-\\frac{2}{9}\\right)^{-2} \\cdot \\left(-\\frac{4}{9}\\right)^3 = \\left(\\frac{9}{2}\\right)^2 \\cdot \\left(-\\frac{4}{9}\\right)^3 = \\left(\\frac{3^{2}}{2}\\right)^2 \\cdot \\left(-\\frac{2^{2}}{3^{2}}\\right)^3 = \\frac{3^{4}}{2^{2}} \\cdot -\\frac{2^{6}}{3^{6}} = -\\frac{2^{4}}{3^{2}} = -\\frac{16}{9}$"], ["$\\displaystyle \\left(\\frac{1}{2}+\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{2}+\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3} = \\left(\\frac{5}{10}+\\frac{1}{10}\\right)^{-2} \\cdot \\left(-\\frac{5}{6}\\right)^{-3} = \\left(\\frac{3}{5}\\right)^{-2} \\cdot \\left(-\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{3}\\right)^2 \\cdot \\left(-\\frac{6}{5}\\right)^3 = \\left(\\frac{5}{3}\\right)^2 \\cdot \\left(-\\frac{2 \\cdot 3}{5}\\right)^3 = \\frac{5^{2}}{3^{2}} \\cdot -\\frac{2^{3} \\cdot 3^{3}}{5^{3}} = -\\frac{2^{3} \\cdot 3}{5} = -\\frac{24}{5}$"], ["$\\displaystyle \\left(-\\frac{3}{8}-\\frac{11}{24}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3}$", "$\\displaystyle \\left(-\\frac{3}{8}-\\frac{11}{24}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3} = \\left(-\\frac{9}{24}-\\frac{11}{24}\\right)^{-2} \\cdot \\left(\\frac{4}{5}\\right)^{-3} = \\left(-\\frac{5}{6}\\right)^{-2} \\cdot \\left(\\frac{5}{4}\\right)^3 = \\left(\\frac{6}{5}\\right)^2 \\cdot \\left(\\frac{5}{4}\\right)^3 = \\left(\\frac{2 \\cdot 3}{5}\\right)^2 \\cdot \\left(\\frac{5}{2^{2}}\\right)^3 = \\frac{2^{2} \\cdot 3^{2}}{5^{2}} \\cdot \\frac{5^{3}}{2^{6}} = \\frac{3^{2} \\cdot 5}{2^{4}} = \\frac{45}{16}$"], ["$\\displaystyle \\left(\\frac{1}{3}-\\frac{16}{21}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{3}-\\frac{16}{21}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3} = \\left(\\frac{7}{21}-\\frac{16}{21}\\right)^{-2} \\cdot \\left(-\\frac{7}{6}\\right)^{-3} = \\left(-\\frac{3}{7}\\right)^{-2} \\cdot \\left(-\\frac{6}{7}\\right)^3 = \\left(\\frac{7}{3}\\right)^2 \\cdot \\left(-\\frac{6}{7}\\right)^3 = \\left(\\frac{7}{3}\\right)^2 \\cdot \\left(-\\frac{2 \\cdot 3}{7}\\right)^3 = \\frac{7^{2}}{3^{2}} \\cdot -\\frac{2^{3} \\cdot 3^{3}}{7^{3}} = -\\frac{2^{3} \\cdot 3}{7} = -\\frac{24}{7}$"], ["$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{5}\\right)^{-2} \\cdot \\left(\\frac{5}{2}\\right)^{-3}$", "$\\displaystyle \\left(\\frac{1}{2}-\\frac{1}{5}\\right)^{-2} \\cdot \\left(\\frac{5}{2}\\right)^{-3} = \\left(\\frac{5}{10}-\\frac{2}{10}\\right)^{-2} \\cdot \\left(\\frac{5}{2}\\right)^{-3} = \\left(\\frac{3}{10}\\right)^{-2} \\cdot \\left(\\frac{2}{5}\\right)^3 = \\left(\\frac{10}{3}\\right)^2 \\cdot \\left(\\frac{2}{5}\\right)^3 = \\left(\\frac{2 \\cdot 5}{3}\\right)^2 \\cdot \\left(\\frac{2}{5}\\right)^3 = \\frac{2^{2} \\cdot 5^{2}}{3^{2}} \\cdot \\frac{2^{3}}{5^{3}} = \\frac{2^{5}}{3^{2} \\cdot 5} = \\frac{32}{45}$"]],
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