miniaufgabe.js
==== 20. September 2021 bis 24. September 2021 ====
=== Donnerstag 23. September 2021 ===
Ausrechnen, Resultat als gekürzter Bruch:miniAufgabe("#exonumbercrunch2","#solnumbercrunch2",
[["$\\displaystyle \\frac{\\frac{3}{\\frac{9}{5}}+\\frac{-\\frac{7}{2}}{-7}}{\\frac{-2}{\\frac{24}{7}}+\\frac{-\\frac{13}{6}}{2}}$", "$\\displaystyle \\frac{\\frac{3}{\\frac{9}{5}}+\\frac{-\\frac{7}{2}}{-7}}{\\frac{-2}{\\frac{24}{7}}+\\frac{-\\frac{13}{6}}{2}} = \\frac{3\\cdot\\frac{5}{9}+-\\frac{7}{2}\\cdot-\\frac{1}{7}}{-2\\cdot\\frac{7}{24}+-\\frac{13}{6}\\cdot\\frac{1}{2}} = \\frac{\\frac{5}{3}+\\frac{1}{2}}{-\\frac{7}{12}-\\frac{13}{12}} = \\frac{\\frac{10}{6}+\\frac{3}{6}}{-\\frac{7}{12}-\\frac{13}{12}} = \\frac{\\frac{13}{6}}{-\\frac{5}{3}} = \\frac{13}{6} \\cdot -\\frac{3}{5} = -\\frac{13}{10}$"], ["$\\displaystyle \\frac{\\frac{-2}{\\frac{9}{5}}+\\frac{\\frac{16}{9}}{4}}{\\frac{3}{-\\frac{13}{2}}+\\frac{\\frac{12}{13}}{-3}}$", "$\\displaystyle \\frac{\\frac{-2}{\\frac{9}{5}}+\\frac{\\frac{16}{9}}{4}}{\\frac{3}{-\\frac{13}{2}}+\\frac{\\frac{12}{13}}{-3}} = \\frac{-2\\cdot\\frac{5}{9}+\\frac{16}{9}\\cdot\\frac{1}{4}}{3\\cdot-\\frac{2}{13}+\\frac{12}{13}\\cdot-\\frac{1}{3}} = \\frac{-\\frac{10}{9}+\\frac{4}{9}}{-\\frac{6}{13}-\\frac{4}{13}} = \\frac{-\\frac{10}{9}+\\frac{4}{9}}{-\\frac{6}{13}-\\frac{4}{13}} = \\frac{-\\frac{2}{3}}{-\\frac{10}{13}} = -\\frac{2}{3} \\cdot -\\frac{13}{10} = \\frac{13}{15}$"], ["$\\displaystyle \\frac{\\frac{2}{\\frac{24}{17}}+\\frac{-\\frac{20}{3}}{4}}{\\frac{2}{-\\frac{12}{5}}+\\frac{\\frac{5}{3}}{-5}}$", "$\\displaystyle \\frac{\\frac{2}{\\frac{24}{17}}+\\frac{-\\frac{20}{3}}{4}}{\\frac{2}{-\\frac{12}{5}}+\\frac{\\frac{5}{3}}{-5}} = \\frac{2\\cdot\\frac{17}{24}+-\\frac{20}{3}\\cdot\\frac{1}{4}}{2\\cdot-\\frac{5}{12}+\\frac{5}{3}\\cdot-\\frac{1}{5}} = \\frac{\\frac{17}{12}-\\frac{5}{3}}{-\\frac{5}{6}-\\frac{1}{3}} = \\frac{\\frac{17}{12}-\\frac{20}{12}}{-\\frac{5}{6}-\\frac{2}{6}} = \\frac{-\\frac{1}{4}}{-\\frac{7}{6}} = -\\frac{1}{4} \\cdot -\\frac{6}{7} = \\frac{3}{14}$"], ["$\\displaystyle \\frac{\\frac{6}{-\\frac{20}{3}}+\\frac{\\frac{5}{2}}{3}}{\\frac{-5}{\\frac{15}{4}}+\\frac{-\\frac{10}{3}}{-3}}$", "$\\displaystyle \\frac{\\frac{6}{-\\frac{20}{3}}+\\frac{\\frac{5}{2}}{3}}{\\frac{-5}{\\frac{15}{4}}+\\frac{-\\frac{10}{3}}{-3}} = \\frac{6\\cdot-\\frac{3}{20}+\\frac{5}{2}\\cdot\\frac{1}{3}}{-5\\cdot\\frac{4}{15}+-\\frac{10}{3}\\cdot-\\frac{1}{3}} = \\frac{-\\frac{9}{10}+\\frac{5}{6}}{-\\frac{4}{3}+\\frac{10}{9}} = \\frac{-\\frac{27}{30}+\\frac{25}{30}}{-\\frac{12}{9}+\\frac{10}{9}} = \\frac{-\\frac{1}{15}}{-\\frac{2}{9}} = -\\frac{1}{15} \\cdot -\\frac{9}{2} = \\frac{3}{10}$"], ["$\\displaystyle \\frac{\\frac{2}{\\frac{20}{13}}+\\frac{-\\frac{18}{5}}{6}}{\\frac{2}{-\\frac{16}{5}}+\\frac{-\\frac{5}{2}}{10}}$", "$\\displaystyle \\frac{\\frac{2}{\\frac{20}{13}}+\\frac{-\\frac{18}{5}}{6}}{\\frac{2}{-\\frac{16}{5}}+\\frac{-\\frac{5}{2}}{10}} = \\frac{2\\cdot\\frac{13}{20}+-\\frac{18}{5}\\cdot\\frac{1}{6}}{2\\cdot-\\frac{5}{16}+-\\frac{5}{2}\\cdot\\frac{1}{10}} = \\frac{\\frac{13}{10}-\\frac{3}{5}}{-\\frac{5}{8}-\\frac{1}{4}} = \\frac{\\frac{13}{10}-\\frac{6}{10}}{-\\frac{5}{8}-\\frac{2}{8}} = \\frac{\\frac{7}{10}}{-\\frac{7}{8}} = \\frac{7}{10} \\cdot -\\frac{8}{7} = -\\frac{4}{5}$"], ["$\\displaystyle \\frac{\\frac{2}{\\frac{20}{7}}+\\frac{\\frac{9}{2}}{-3}}{\\frac{-8}{-\\frac{14}{3}}+\\frac{-\\frac{16}{7}}{2}}$", "$\\displaystyle \\frac{\\frac{2}{\\frac{20}{7}}+\\frac{\\frac{9}{2}}{-3}}{\\frac{-8}{-\\frac{14}{3}}+\\frac{-\\frac{16}{7}}{2}} = \\frac{2\\cdot\\frac{7}{20}+\\frac{9}{2}\\cdot-\\frac{1}{3}}{-8\\cdot-\\frac{3}{14}+-\\frac{16}{7}\\cdot\\frac{1}{2}} = \\frac{\\frac{7}{10}-\\frac{3}{2}}{\\frac{12}{7}-\\frac{8}{7}} = \\frac{\\frac{7}{10}-\\frac{15}{10}}{\\frac{12}{7}-\\frac{8}{7}} = \\frac{-\\frac{4}{5}}{\\frac{4}{7}} = -\\frac{4}{5} \\cdot \\frac{7}{4} = -\\frac{7}{5}$"], ["$\\displaystyle \\frac{\\frac{-2}{-\\frac{8}{5}}+\\frac{-\\frac{17}{6}}{2}}{\\frac{-2}{\\frac{24}{7}}+\\frac{\\frac{8}{3}}{-8}}$", "$\\displaystyle \\frac{\\frac{-2}{-\\frac{8}{5}}+\\frac{-\\frac{17}{6}}{2}}{\\frac{-2}{\\frac{24}{7}}+\\frac{\\frac{8}{3}}{-8}} = \\frac{-2\\cdot-\\frac{5}{8}+-\\frac{17}{6}\\cdot\\frac{1}{2}}{-2\\cdot\\frac{7}{24}+\\frac{8}{3}\\cdot-\\frac{1}{8}} = \\frac{\\frac{5}{4}-\\frac{17}{12}}{-\\frac{7}{12}-\\frac{1}{3}} = \\frac{\\frac{15}{12}-\\frac{17}{12}}{-\\frac{7}{12}-\\frac{4}{12}} = \\frac{-\\frac{1}{6}}{-\\frac{11}{12}} = -\\frac{1}{6} \\cdot -\\frac{12}{11} = \\frac{2}{11}$"], ["$\\displaystyle \\frac{\\frac{2}{-\\frac{13}{7}}+\\frac{-\\frac{14}{13}}{-2}}{\\frac{-6}{-\\frac{13}{2}}+\\frac{-\\frac{18}{13}}{9}}$", "$\\displaystyle \\frac{\\frac{2}{-\\frac{13}{7}}+\\frac{-\\frac{14}{13}}{-2}}{\\frac{-6}{-\\frac{13}{2}}+\\frac{-\\frac{18}{13}}{9}} = \\frac{2\\cdot-\\frac{7}{13}+-\\frac{14}{13}\\cdot-\\frac{1}{2}}{-6\\cdot-\\frac{2}{13}+-\\frac{18}{13}\\cdot\\frac{1}{9}} = \\frac{-\\frac{14}{13}+\\frac{7}{13}}{\\frac{12}{13}-\\frac{2}{13}} = \\frac{-\\frac{14}{13}+\\frac{7}{13}}{\\frac{12}{13}-\\frac{2}{13}} = \\frac{-\\frac{7}{13}}{\\frac{10}{13}} = -\\frac{7}{13} \\cdot \\frac{13}{10} = -\\frac{7}{10}$"], ["$\\displaystyle \\frac{\\frac{-2}{-\\frac{11}{2}}+\\frac{\\frac{16}{11}}{2}}{\\frac{2}{\\frac{22}{5}}+\\frac{\\frac{30}{11}}{-2}}$", "$\\displaystyle \\frac{\\frac{-2}{-\\frac{11}{2}}+\\frac{\\frac{16}{11}}{2}}{\\frac{2}{\\frac{22}{5}}+\\frac{\\frac{30}{11}}{-2}} = \\frac{-2\\cdot-\\frac{2}{11}+\\frac{16}{11}\\cdot\\frac{1}{2}}{2\\cdot\\frac{5}{22}+\\frac{30}{11}\\cdot-\\frac{1}{2}} = \\frac{\\frac{4}{11}+\\frac{8}{11}}{\\frac{5}{11}-\\frac{15}{11}} = \\frac{\\frac{4}{11}+\\frac{8}{11}}{\\frac{5}{11}-\\frac{15}{11}} = \\frac{\\frac{12}{11}}{-\\frac{10}{11}} = \\frac{12}{11} \\cdot -\\frac{11}{10} = -\\frac{6}{5}$"], ["$\\displaystyle \\frac{\\frac{4}{-\\frac{11}{2}}+\\frac{\\frac{12}{11}}{-2}}{\\frac{2}{-\\frac{16}{19}}+\\frac{-\\frac{15}{8}}{-3}}$", "$\\displaystyle \\frac{\\frac{4}{-\\frac{11}{2}}+\\frac{\\frac{12}{11}}{-2}}{\\frac{2}{-\\frac{16}{19}}+\\frac{-\\frac{15}{8}}{-3}} = \\frac{4\\cdot-\\frac{2}{11}+\\frac{12}{11}\\cdot-\\frac{1}{2}}{2\\cdot-\\frac{19}{16}+-\\frac{15}{8}\\cdot-\\frac{1}{3}} = \\frac{-\\frac{8}{11}-\\frac{6}{11}}{-\\frac{19}{8}+\\frac{5}{8}} = \\frac{-\\frac{8}{11}-\\frac{6}{11}}{-\\frac{19}{8}+\\frac{5}{8}} = \\frac{-\\frac{14}{11}}{-\\frac{7}{4}} = -\\frac{14}{11} \\cdot -\\frac{4}{7} = \\frac{8}{11}$"]],
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=== Freitag 24. September 2021 ===
Primfaktorzerlegung: Schreiben Sie als Produkt von Potenzen von Primzahlen, nach aufsteigender Basis geordnet.miniAufgabe("#exoprimfaktorzerlegung","#solprimfaktorzerlegung",
[["$84 \\cdot 2310 \\cdot 1100$", "$84 \\cdot 2310 \\cdot 1100 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot 10 \\cdot 11 \\,\\, \\cdot \\,\\, 10^{2} \\cdot 11 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot (2 \\cdot 5) \\cdot 11 \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{2} \\cdot 11 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 5^{2} \\cdot 11 \\quad = \\quad 2^{5} \\cdot 3^{2} \\cdot 7^{2} \\cdot 11^{2}$"], ["$735 \\cdot 231000 \\cdot 2900$", "$735 \\cdot 231000 \\cdot 2900 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot 10^{3} \\cdot 11 \\,\\, \\cdot \\,\\, 10^{2} \\cdot 29 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot (2 \\cdot 5)^{3} \\cdot 11 \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{2} \\cdot 29 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2^{3} \\cdot 3 \\cdot 5^{3} \\cdot 7 \\cdot 11 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 5^{2} \\cdot 29 \\quad = \\quad 2^{5} \\cdot 3^{2} \\cdot 5^{6} \\cdot 7^{3} \\cdot 11 \\cdot 29$"], ["$315 \\cdot 1650 \\cdot 290000$", "$315 \\cdot 1650 \\cdot 290000 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 10 \\cdot 11 \\,\\, \\cdot \\,\\, 10^{4} \\cdot 29 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot (2 \\cdot 5) \\cdot 11 \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{4} \\cdot 29 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 5^{2} \\cdot 11 \\,\\, \\cdot \\,\\, 2^{4} \\cdot 5^{4} \\cdot 29 \\quad = \\quad 2^{5} \\cdot 3^{3} \\cdot 5^{7} \\cdot 7 \\cdot 11 \\cdot 29$"], ["$36 \\cdot 105000 \\cdot 130000$", "$36 \\cdot 105000 \\cdot 130000 \\quad = \\quad 2^{2} \\cdot 3^{2} \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 7 \\cdot 10^{3} \\,\\, \\cdot \\,\\, 10^{4} \\cdot 13 \\quad = \\quad 2^{2} \\cdot 3^{2} \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 7 \\cdot (2 \\cdot 5)^{3} \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{4} \\cdot 13 \\quad = \\quad 2^{2} \\cdot 3^{2} \\,\\, \\cdot \\,\\, 2^{3} \\cdot 3 \\cdot 5^{4} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{4} \\cdot 5^{4} \\cdot 13 \\quad = \\quad 2^{9} \\cdot 3^{3} \\cdot 5^{8} \\cdot 7 \\cdot 13$"], ["$1225 \\cdot 4200 \\cdot 23000$", "$1225 \\cdot 4200 \\cdot 23000 \\quad = \\quad 5^{2} \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 7 \\cdot 10^{2} \\,\\, \\cdot \\,\\, 10^{3} \\cdot 23 \\quad = \\quad 5^{2} \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 7 \\cdot (2 \\cdot 5)^{2} \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{3} \\cdot 23 \\quad = \\quad 5^{2} \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2^{3} \\cdot 3 \\cdot 5^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{3} \\cdot 5^{3} \\cdot 23 \\quad = \\quad 2^{6} \\cdot 3 \\cdot 5^{7} \\cdot 7^{3} \\cdot 23$"], ["$315 \\cdot 6300 \\cdot 1700$", "$315 \\cdot 6300 \\cdot 1700 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3^{2} \\cdot 7 \\cdot 10^{2} \\,\\, \\cdot \\,\\, 10^{2} \\cdot 17 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3^{2} \\cdot 7 \\cdot (2 \\cdot 5)^{2} \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{2} \\cdot 17 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 3^{2} \\cdot 5^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 5^{2} \\cdot 17 \\quad = \\quad 2^{4} \\cdot 3^{4} \\cdot 5^{5} \\cdot 7^{2} \\cdot 17$"], ["$315 \\cdot 23100 \\cdot 290000$", "$315 \\cdot 23100 \\cdot 290000 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot 10^{2} \\cdot 11 \\,\\, \\cdot \\,\\, 10^{4} \\cdot 29 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 7 \\cdot (2 \\cdot 5)^{2} \\cdot 11 \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{4} \\cdot 29 \\quad = \\quad 3^{2} \\cdot 5 \\cdot 7 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 3 \\cdot 5^{2} \\cdot 7 \\cdot 11 \\,\\, \\cdot \\,\\, 2^{4} \\cdot 5^{4} \\cdot 29 \\quad = \\quad 2^{6} \\cdot 3^{3} \\cdot 5^{7} \\cdot 7^{2} \\cdot 11 \\cdot 29$"], ["$84 \\cdot 16500 \\cdot 1100$", "$84 \\cdot 16500 \\cdot 1100 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 10^{2} \\cdot 11 \\,\\, \\cdot \\,\\, 10^{2} \\cdot 11 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot (2 \\cdot 5)^{2} \\cdot 11 \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{2} \\cdot 11 \\quad = \\quad 2^{2} \\cdot 3 \\cdot 7 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 3 \\cdot 5^{3} \\cdot 11 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 5^{2} \\cdot 11 \\quad = \\quad 2^{6} \\cdot 3^{2} \\cdot 5^{5} \\cdot 7 \\cdot 11^{2}$"], ["$126 \\cdot 10500 \\cdot 7000$", "$126 \\cdot 10500 \\cdot 7000 \\quad = \\quad 2 \\cdot 3^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 7 \\cdot 10^{2} \\,\\, \\cdot \\,\\, 7 \\cdot 10^{3} \\quad = \\quad 2 \\cdot 3^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 3 \\cdot 5 \\cdot 7 \\cdot (2 \\cdot 5)^{2} \\,\\, \\cdot \\,\\, 7 \\cdot (2 \\cdot 5)^{3} \\quad = \\quad 2 \\cdot 3^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{2} \\cdot 3 \\cdot 5^{3} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{3} \\cdot 5^{3} \\cdot 7 \\quad = \\quad 2^{6} \\cdot 3^{3} \\cdot 5^{6} \\cdot 7^{3}$"], ["$735 \\cdot 4200 \\cdot 13000$", "$735 \\cdot 4200 \\cdot 13000 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 7 \\cdot 10^{2} \\,\\, \\cdot \\,\\, 10^{3} \\cdot 13 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2 \\cdot 3 \\cdot 7 \\cdot (2 \\cdot 5)^{2} \\,\\, \\cdot \\,\\, (2 \\cdot 5)^{3} \\cdot 13 \\quad = \\quad 3 \\cdot 5 \\cdot 7^{2} \\,\\, \\cdot \\,\\, 2^{3} \\cdot 3 \\cdot 5^{2} \\cdot 7 \\,\\, \\cdot \\,\\, 2^{3} \\cdot 5^{3} \\cdot 13 \\quad = \\quad 2^{6} \\cdot 3^{2} \\cdot 5^{6} \\cdot 7^{3} \\cdot 13$"]],
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