miniaufgabe.js
==== 13. September 2021 bis 17. September 2021 ====
=== Donnerstag 16. September 2021 ===
Potenzgesetze anwenden, kürzen, am Schluss als einen einfachen Bruch (bzw. natürliche Zahl) schreiben:miniAufgabe("#exonumbercrunch1","#solnumbercrunch1",
[["$\\displaystyle \\frac{5 \\cdot 25^4 \\cdot 16 \\cdot 11 \\cdot 121^2}{25 \\cdot 125^2 \\cdot 32 \\cdot 121^2}$", "$\\displaystyle \\frac{5 \\cdot 25^4 \\cdot 16 \\cdot 11 \\cdot 121^2}{25 \\cdot 125^2 \\cdot 32 \\cdot 121^2} = \\frac{5 \\cdot \\left(5^{2}\\right)^{4} \\cdot 2^{4} \\cdot 11 \\cdot \\left(11^{2}\\right)^{2}}{5^2 \\cdot \\left(5^{3}\\right)^{2} \\cdot 2^{5} \\cdot \\left(11^{2}\\right)^{2}} = \\frac{5^{9} \\cdot 2^{4} \\cdot 11^{5}}{5^{8} \\cdot 2^{5} \\cdot 11^{4}} = \\frac{5 \\cdot 11}{2} = \\frac{55}{2}$"], ["$\\displaystyle \\frac{125 \\cdot 27 \\cdot 81 \\cdot 49^6}{25^2 \\cdot 9 \\cdot 27^2 \\cdot 7 \\cdot 49^6}$", "$\\displaystyle \\frac{125 \\cdot 27 \\cdot 81 \\cdot 49^6}{25^2 \\cdot 9 \\cdot 27^2 \\cdot 7 \\cdot 49^6} = \\frac{5^{3} \\cdot 3^3 \\cdot 3^{4} \\cdot \\left(7^{2}\\right)^{6}}{\\left(5^{2}\\right)^{2} \\cdot 3^2 \\cdot \\left(3^{3}\\right)^{2} \\cdot 7 \\cdot \\left(7^{2}\\right)^{6}} = \\frac{5^{3} \\cdot 3^{7} \\cdot 7^{12}}{5^{4} \\cdot 3^{8} \\cdot 7^{13}} = \\frac{1}{5 \\cdot 3 \\cdot 7} = \\frac{1}{105}$"], ["$\\displaystyle \\frac{32 \\cdot 64 \\cdot 125^4 \\cdot 7 \\cdot 49^4}{4 \\cdot 1024 \\cdot 5 \\cdot 25^6 \\cdot 49^4}$", "$\\displaystyle \\frac{32 \\cdot 64 \\cdot 125^4 \\cdot 7 \\cdot 49^4}{4 \\cdot 1024 \\cdot 5 \\cdot 25^6 \\cdot 49^4} = \\frac{2^5 \\cdot 2^{6} \\cdot \\left(5^{3}\\right)^{4} \\cdot 7 \\cdot \\left(7^{2}\\right)^{4}}{2^2 \\cdot 2^{10} \\cdot 5 \\cdot \\left(5^{2}\\right)^{6} \\cdot \\left(7^{2}\\right)^{4}} = \\frac{2^{11} \\cdot 5^{12} \\cdot 7^{9}}{2^{12} \\cdot 5^{13} \\cdot 7^{8}} = \\frac{7}{2 \\cdot 5} = \\frac{7}{10}$"], ["$\\displaystyle \\frac{5 \\cdot 125^4 \\cdot 3 \\cdot 9^3 \\cdot 11 \\cdot 121^3}{25^7 \\cdot 9^4 \\cdot 121^3}$", "$\\displaystyle \\frac{5 \\cdot 125^4 \\cdot 3 \\cdot 9^3 \\cdot 11 \\cdot 121^3}{25^7 \\cdot 9^4 \\cdot 121^3} = \\frac{5 \\cdot \\left(5^{3}\\right)^{4} \\cdot 3 \\cdot \\left(3^{2}\\right)^{3} \\cdot 11 \\cdot \\left(11^{2}\\right)^{3}}{\\left(5^{2}\\right)^{7} \\cdot \\left(3^{2}\\right)^{4} \\cdot \\left(11^{2}\\right)^{3}} = \\frac{5^{13} \\cdot 3^{7} \\cdot 11^{7}}{5^{14} \\cdot 3^{8} \\cdot 11^{6}} = \\frac{11}{5 \\cdot 3} = \\frac{11}{15}$"], ["$\\displaystyle \\frac{5 \\cdot 25 \\cdot 64 \\cdot 11 \\cdot 121^4}{25 \\cdot 2 \\cdot 4^3 \\cdot 121^4}$", "$\\displaystyle \\frac{5 \\cdot 25 \\cdot 64 \\cdot 11 \\cdot 121^4}{25 \\cdot 2 \\cdot 4^3 \\cdot 121^4} = \\frac{5 \\cdot 5^{2} \\cdot 2^{6} \\cdot 11 \\cdot \\left(11^{2}\\right)^{4}}{5^{2} \\cdot 2 \\cdot \\left(2^{2}\\right)^{3} \\cdot \\left(11^{2}\\right)^{4}} = \\frac{5^{3} \\cdot 2^{6} \\cdot 11^{9}}{5^{2} \\cdot 2^{7} \\cdot 11^{8}} = \\frac{5 \\cdot 11}{2} = \\frac{55}{2}$"], ["$\\displaystyle \\frac{16 \\cdot 32^2 \\cdot 25 \\cdot 125 \\cdot 121^4}{2 \\cdot 16^3 \\cdot 125^2 \\cdot 11 \\cdot 121^3}$", "$\\displaystyle \\frac{16 \\cdot 32^2 \\cdot 25 \\cdot 125 \\cdot 121^4}{2 \\cdot 16^3 \\cdot 125^2 \\cdot 11 \\cdot 121^3} = \\frac{2^4 \\cdot \\left(2^{5}\\right)^{2} \\cdot 5^2 \\cdot 5^{3} \\cdot \\left(11^{2}\\right)^{4}}{2 \\cdot \\left(2^{4}\\right)^{3} \\cdot \\left(5^{3}\\right)^{2} \\cdot 11 \\cdot \\left(11^{2}\\right)^{3}} = \\frac{2^{14} \\cdot 5^{5} \\cdot 11^{8}}{2^{13} \\cdot 5^{6} \\cdot 11^{7}} = \\frac{2 \\cdot 11}{5} = \\frac{22}{5}$"], ["$\\displaystyle \\frac{81^4 \\cdot 4 \\cdot 8^3 \\cdot 7 \\cdot 49^2}{3 \\cdot 9^8 \\cdot 4^5 \\cdot 49^2}$", "$\\displaystyle \\frac{81^4 \\cdot 4 \\cdot 8^3 \\cdot 7 \\cdot 49^2}{3 \\cdot 9^8 \\cdot 4^5 \\cdot 49^2} = \\frac{\\left(3^{4}\\right)^{4} \\cdot 2^2 \\cdot \\left(2^{3}\\right)^{3} \\cdot 7 \\cdot \\left(7^{2}\\right)^{2}}{3 \\cdot \\left(3^{2}\\right)^{8} \\cdot \\left(2^{2}\\right)^{5} \\cdot \\left(7^{2}\\right)^{2}} = \\frac{3^{16} \\cdot 2^{11} \\cdot 7^{5}}{3^{17} \\cdot 2^{10} \\cdot 7^{4}} = \\frac{2 \\cdot 7}{3} = \\frac{14}{3}$"], ["$\\displaystyle \\frac{16 \\cdot 32^2 \\cdot 9^6 \\cdot 121^8}{2 \\cdot 128^2 \\cdot 27 \\cdot 81^2 \\cdot 11 \\cdot 121^7}$", "$\\displaystyle \\frac{16 \\cdot 32^2 \\cdot 9^6 \\cdot 121^8}{2 \\cdot 128^2 \\cdot 27 \\cdot 81^2 \\cdot 11 \\cdot 121^7} = \\frac{2^4 \\cdot \\left(2^{5}\\right)^{2} \\cdot \\left(3^{2}\\right)^{6} \\cdot \\left(11^{2}\\right)^{8}}{2 \\cdot \\left(2^{7}\\right)^{2} \\cdot 3^3 \\cdot \\left(3^{4}\\right)^{2} \\cdot 11 \\cdot \\left(11^{2}\\right)^{7}} = \\frac{2^{14} \\cdot 3^{12} \\cdot 11^{16}}{2^{15} \\cdot 3^{11} \\cdot 11^{15}} = \\frac{3 \\cdot 11}{2} = \\frac{33}{2}$"], ["$\\displaystyle \\frac{4 \\cdot 3 \\cdot 27 \\cdot 13 \\cdot 169^2}{2 \\cdot 3 \\cdot 9 \\cdot 169^3}$", "$\\displaystyle \\frac{4 \\cdot 3 \\cdot 27 \\cdot 13 \\cdot 169^2}{2 \\cdot 3 \\cdot 9 \\cdot 169^3} = \\frac{2^{2} \\cdot 3 \\cdot 3^{3} \\cdot 13 \\cdot \\left(13^{2}\\right)^{2}}{2 \\cdot 3 \\cdot 3^{2} \\cdot \\left(13^{2}\\right)^{3}} = \\frac{2^{2} \\cdot 3^{4} \\cdot 13^{5}}{2 \\cdot 3^{3} \\cdot 13^{6}} = \\frac{2 \\cdot 3}{13} = \\frac{6}{13}$"], ["$\\displaystyle \\frac{9 \\cdot 25 \\cdot 125^2 \\cdot 169^8}{27 \\cdot 5 \\cdot 125^2 \\cdot 13 \\cdot 169^8}$", "$\\displaystyle \\frac{9 \\cdot 25 \\cdot 125^2 \\cdot 169^8}{27 \\cdot 5 \\cdot 125^2 \\cdot 13 \\cdot 169^8} = \\frac{3^{2} \\cdot 5^2 \\cdot \\left(5^{3}\\right)^{2} \\cdot \\left(13^{2}\\right)^{8}}{3^{3} \\cdot 5 \\cdot \\left(5^{3}\\right)^{2} \\cdot 13 \\cdot \\left(13^{2}\\right)^{8}} = \\frac{3^{2} \\cdot 5^{8} \\cdot 13^{16}}{3^{3} \\cdot 5^{7} \\cdot 13^{17}} = \\frac{5}{3 \\cdot 13} = \\frac{5}{39}$"]],
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=== Freitag 17. September 2021 ===
Lösen Sie die Gleichung nach $x$ auf.miniAufgabe("#exolinGleich2","#sollinGleich2",
[["$\\displaystyle \\frac{9}{8}\\cdot \\left(-\\frac{16}{5}-\\frac{8}{3}x\\right) = -\\frac{9}{13} \\cdot \\left(-\\frac{91}{45}+\\frac{26}{3}x\\right)$", "$$\\begin{align*}\n\\frac{9}{8}\\cdot \\left(-\\frac{16}{5}-\\frac{8}{3}x\\right) & = -\\frac{9}{13} \\cdot \\left(-\\frac{91}{45}+\\frac{26}{3}x\\right) && |\\text{TU}\\\\\n-\\frac{18}{5}-3x & = \\frac{7}{5}-6x && |+\\frac{18}{5}\\\\\n-3x & = 5-6x && |+6x\\\\\n3x & = 5 && |: 3\\\\\nx & = \\frac{5}{3}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{4}{3}\\cdot \\left(-\\frac{15}{16}-\\frac{15}{2}x\\right) = -\\frac{6}{5} \\cdot \\left(\\frac{55}{8}+\\frac{25}{2}x\\right)$", "$$\\begin{align*}\n\\frac{4}{3}\\cdot \\left(-\\frac{15}{16}-\\frac{15}{2}x\\right) & = -\\frac{6}{5} \\cdot \\left(\\frac{55}{8}+\\frac{25}{2}x\\right) && |\\text{TU}\\\\\n-\\frac{5}{4}-10x & = -\\frac{33}{4}-15x && |+\\frac{5}{4}\\\\\n-10x & = -7-15x && |+15x\\\\\n5x & = -7 && |: 5\\\\\nx & = -\\frac{7}{5}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{11}{6}\\cdot \\left(\\frac{16}{11}+\\frac{108}{11}x\\right) = -\\frac{13}{7} \\cdot \\left(\\frac{49}{39}-\\frac{98}{13}x\\right)$", "$$\\begin{align*}\n\\frac{11}{6}\\cdot \\left(\\frac{16}{11}+\\frac{108}{11}x\\right) & = -\\frac{13}{7} \\cdot \\left(\\frac{49}{39}-\\frac{98}{13}x\\right) && |\\text{TU}\\\\\n\\frac{8}{3}+18x & = -\\frac{7}{3}+14x && |-\\frac{8}{3}\\\\\n18x & = -5+14x && |-14x\\\\\n4x & = -5 && |: 4\\\\\nx & = -\\frac{5}{4}\n\\end{align*}\n$$"], ["$\\displaystyle -\\frac{7}{5}\\cdot \\left(-\\frac{85}{42}-\\frac{80}{7}x\\right) = \\frac{9}{4} \\cdot \\left(\\frac{94}{27}+\\frac{28}{9}x\\right)$", "$$\\begin{align*}\n-\\frac{7}{5}\\cdot \\left(-\\frac{85}{42}-\\frac{80}{7}x\\right) & = \\frac{9}{4} \\cdot \\left(\\frac{94}{27}+\\frac{28}{9}x\\right) && |\\text{TU}\\\\\n\\frac{17}{6}+16x & = \\frac{47}{6}+7x && |-\\frac{17}{6}\\\\\n16x & = 5+7x && |-7x\\\\\n9x & = 5 && |: 9\\\\\nx & = \\frac{5}{9}\n\\end{align*}\n$$"], ["$\\displaystyle -\\frac{4}{3}\\cdot \\left(\\frac{15}{16}+\\frac{3}{4}x\\right) = -\\frac{9}{11} \\cdot \\left(-\\frac{33}{4}+\\frac{88}{9}x\\right)$", "$$\\begin{align*}\n-\\frac{4}{3}\\cdot \\left(\\frac{15}{16}+\\frac{3}{4}x\\right) & = -\\frac{9}{11} \\cdot \\left(-\\frac{33}{4}+\\frac{88}{9}x\\right) && |\\text{TU}\\\\\n-\\frac{5}{4}-x & = \\frac{27}{4}-8x && |+\\frac{5}{4}\\\\\n-x & = 8-8x && |+8x\\\\\n7x & = 8 && |: 7\\\\\nx & = \\frac{8}{7}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{2}{3}\\cdot \\left(7+\\frac{45}{2}x\\right) = \\frac{5}{9} \\cdot \\left(39+\\frac{72}{5}x\\right)$", "$$\\begin{align*}\n\\frac{2}{3}\\cdot \\left(7+\\frac{45}{2}x\\right) & = \\frac{5}{9} \\cdot \\left(39+\\frac{72}{5}x\\right) && |\\text{TU}\\\\\n\\frac{14}{3}+15x & = \\frac{65}{3}+8x && |-\\frac{14}{3}\\\\\n15x & = 17+8x && |-8x\\\\\n7x & = 17 && |: 7\\\\\nx & = \\frac{17}{7}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{7}{3}\\cdot \\left(-\\frac{13}{21}-\\frac{27}{7}x\\right) = \\frac{5}{6} \\cdot \\left(-\\frac{52}{3}-18x\\right)$", "$$\\begin{align*}\n\\frac{7}{3}\\cdot \\left(-\\frac{13}{21}-\\frac{27}{7}x\\right) & = \\frac{5}{6} \\cdot \\left(-\\frac{52}{3}-18x\\right) && |\\text{TU}\\\\\n-\\frac{13}{9}-9x & = -\\frac{130}{9}-15x && |+\\frac{13}{9}\\\\\n-9x & = -13-15x && |+15x\\\\\n6x & = -13 && |: 6\\\\\nx & = -\\frac{13}{6}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{13}{12}\\cdot \\left(\\frac{64}{13}-\\frac{72}{13}x\\right) = \\frac{13}{8} \\cdot \\left(-\\frac{40}{39}-\\frac{120}{13}x\\right)$", "$$\\begin{align*}\n\\frac{13}{12}\\cdot \\left(\\frac{64}{13}-\\frac{72}{13}x\\right) & = \\frac{13}{8} \\cdot \\left(-\\frac{40}{39}-\\frac{120}{13}x\\right) && |\\text{TU}\\\\\n\\frac{16}{3}-6x & = -\\frac{5}{3}-15x && |-\\frac{16}{3}\\\\\n-6x & = -7-15x && |+15x\\\\\n9x & = -7 && |: 9\\\\\nx & = -\\frac{7}{9}\n\\end{align*}\n$$"], ["$\\displaystyle \\frac{15}{13}\\cdot \\left(-\\frac{91}{60}-\\frac{13}{3}x\\right) = \\frac{13}{8} \\cdot \\left(-\\frac{102}{13}-\\frac{120}{13}x\\right)$", "$$\\begin{align*}\n\\frac{15}{13}\\cdot \\left(-\\frac{91}{60}-\\frac{13}{3}x\\right) & = \\frac{13}{8} \\cdot \\left(-\\frac{102}{13}-\\frac{120}{13}x\\right) && |\\text{TU}\\\\\n-\\frac{7}{4}-5x & = -\\frac{51}{4}-15x && |+\\frac{7}{4}\\\\\n-5x & = -11-15x && |+15x\\\\\n10x & = -11 && |: 10\\\\\nx & = -\\frac{11}{10}\n\\end{align*}\n$$"], ["$\\displaystyle -\\frac{3}{5}\\cdot \\left(\\frac{65}{9}-25x\\right) = -\\frac{11}{7} \\cdot \\left(\\frac{133}{33}-\\frac{84}{11}x\\right)$", "$$\\begin{align*}\n-\\frac{3}{5}\\cdot \\left(\\frac{65}{9}-25x\\right) & = -\\frac{11}{7} \\cdot \\left(\\frac{133}{33}-\\frac{84}{11}x\\right) && |\\text{TU}\\\\\n-\\frac{13}{3}+15x & = -\\frac{19}{3}+12x && |+\\frac{13}{3}\\\\\n15x & = -2+12x && |-12x\\\\\n3x & = -2 && |: 3\\\\\nx & = -\\frac{2}{3}\n\\end{align*}\n$$"]],
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