miniaufgabe.js ==== 5. März 2018 bis 9. März 2018 ==== === Dienstag 6. März 2018 === Vereinfachen Sie: miniAufgabe("#exoehochxbrueche","#solehochxbrueche", [["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{1}{3}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{3}}}{\\sqrt[8]{\\mathrm{e}^{-3\\cdot x}}}\\right)^{\\frac{2}{3}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{1}{3}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{3}}}{\\sqrt[8]{\\mathrm{e}^{-3\\cdot x}}}\\right)^{\\frac{2}{3}} = \\left(\\frac{\\mathrm{e}^{-\\frac{1}{3}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{2}{3}\\cdot x}}{\\mathrm{e}^{-\\frac{3}{8}\\cdot x}}\\right)^{\\frac{2}{3}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{1}{3} + \\left(-\\frac{2}{3}\\right) - \\left(-\\frac{3}{8}\\right)\\right)} \\right)^{\\frac{2}{3}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{8}{24} + \\left(-\\frac{16}{24}\\right) - \\left(-\\frac{9}{24}\\right)\\right)} \\right)^{\\frac{2}{3}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{15}{24}\\right)} \\right)^{\\frac{2}{3}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{5}{8}\\right)} \\right)^{\\frac{2}{3}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{5}{8}\\right) \\cdot \\frac{2}{3}} = \\mathrm{e}^{-\\frac{5}{12} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{25}{12}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{3}}}{\\sqrt[4]{\\mathrm{e}^{-7\\cdot x}}}\\right)^{-\\frac{4}{5}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{25}{12}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{3}}}{\\sqrt[4]{\\mathrm{e}^{-7\\cdot x}}}\\right)^{-\\frac{4}{5}} = \\left(\\frac{\\mathrm{e}^{-\\frac{25}{12}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{2}{3}\\cdot x}}{\\mathrm{e}^{-\\frac{7}{4}\\cdot x}}\\right)^{-\\frac{4}{5}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{25}{12} + \\left(-\\frac{2}{3}\\right) - \\left(-\\frac{7}{4}\\right)\\right)} \\right)^{-\\frac{4}{5}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{25}{12} + \\left(-\\frac{8}{12}\\right) - \\left(-\\frac{21}{12}\\right)\\right)} \\right)^{-\\frac{4}{5}} = \\left(\\mathrm{e}^{x \\cdot \\left(-1\\right)} \\right)^{-\\frac{4}{5}} = \\left(\\mathrm{e}^{x \\cdot \\left(-1\\right)} \\right)^{-\\frac{4}{5}} = $\n$\\mathrm{e}^{x \\cdot \\left(-1\\right) \\cdot \\left(-\\frac{4}{5}\\right)} = \\mathrm{e}^{\\frac{4}{5} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{23}{10}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{3}{2}}}{\\sqrt[3]{\\mathrm{e}^{2\\cdot x}}}\\right)^{\\frac{3}{11}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{23}{10}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{3}{2}}}{\\sqrt[3]{\\mathrm{e}^{2\\cdot x}}}\\right)^{\\frac{3}{11}} = \\left(\\frac{\\mathrm{e}^{-\\frac{23}{10}\\cdot x} \\cdot \\mathrm{e}^{\\frac{3}{2}\\cdot x}}{\\mathrm{e}^{\\frac{2}{3}\\cdot x}}\\right)^{\\frac{3}{11}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{23}{10} + \\frac{3}{2} - \\frac{2}{3}\\right)} \\right)^{\\frac{3}{11}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{69}{30} + \\frac{45}{30} - \\frac{20}{30}\\right)} \\right)^{\\frac{3}{11}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{44}{30}\\right)} \\right)^{\\frac{3}{11}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{22}{15}\\right)} \\right)^{\\frac{3}{11}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{22}{15}\\right) \\cdot \\frac{3}{11}} = \\mathrm{e}^{-\\frac{2}{5} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{69}{55}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{11}}}{\\sqrt[5]{\\mathrm{e}^{9\\cdot x}}}\\right)^{\\frac{3}{2}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{69}{55}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{11}}}{\\sqrt[5]{\\mathrm{e}^{9\\cdot x}}}\\right)^{\\frac{3}{2}} = \\left(\\frac{\\mathrm{e}^{\\frac{69}{55}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{2}{11}\\cdot x}}{\\mathrm{e}^{\\frac{9}{5}\\cdot x}}\\right)^{\\frac{3}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(\\frac{69}{55} + \\left(-\\frac{2}{11}\\right) - \\frac{9}{5}\\right)} \\right)^{\\frac{3}{2}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(\\frac{69}{55} + \\left(-\\frac{10}{55}\\right) - \\frac{99}{55}\\right)} \\right)^{\\frac{3}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{40}{55}\\right)} \\right)^{\\frac{3}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{8}{11}\\right)} \\right)^{\\frac{3}{2}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{8}{11}\\right) \\cdot \\frac{3}{2}} = \\mathrm{e}^{-\\frac{12}{11} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{7}{24}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{5}{6}}}{\\sqrt[8]{\\mathrm{e}^{-3\\cdot x}}}\\right)^{-\\frac{8}{11}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{7}{24}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{5}{6}}}{\\sqrt[8]{\\mathrm{e}^{-3\\cdot x}}}\\right)^{-\\frac{8}{11}} = \\left(\\frac{\\mathrm{e}^{-\\frac{7}{24}\\cdot x} \\cdot \\mathrm{e}^{\\frac{5}{6}\\cdot x}}{\\mathrm{e}^{-\\frac{3}{8}\\cdot x}}\\right)^{-\\frac{8}{11}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{7}{24} + \\frac{5}{6} - \\left(-\\frac{3}{8}\\right)\\right)} \\right)^{-\\frac{8}{11}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{7}{24} + \\frac{20}{24} - \\left(-\\frac{9}{24}\\right)\\right)} \\right)^{-\\frac{8}{11}} = \\left(\\mathrm{e}^{x \\cdot \\frac{22}{24}} \\right)^{-\\frac{8}{11}} = \\left(\\mathrm{e}^{x \\cdot \\frac{11}{12}} \\right)^{-\\frac{8}{11}} = $\n$\\mathrm{e}^{x \\cdot \\frac{11}{12} \\cdot \\left(-\\frac{8}{11}\\right)} = \\mathrm{e}^{-\\frac{2}{3} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{15}{2}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{5}{2}}}{\\sqrt[5]{\\mathrm{e}^{2\\cdot x}}}\\right)^{\\frac{4}{9}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{15}{2}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{5}{2}}}{\\sqrt[5]{\\mathrm{e}^{2\\cdot x}}}\\right)^{\\frac{4}{9}} = \\left(\\frac{\\mathrm{e}^{-\\frac{15}{2}\\cdot x} \\cdot \\mathrm{e}^{\\frac{5}{2}\\cdot x}}{\\mathrm{e}^{\\frac{2}{5}\\cdot x}}\\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{15}{2} + \\frac{5}{2} - \\frac{2}{5}\\right)} \\right)^{\\frac{4}{9}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{75}{10} + \\frac{25}{10} - \\frac{4}{10}\\right)} \\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{54}{10}\\right)} \\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{27}{5}\\right)} \\right)^{\\frac{4}{9}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{27}{5}\\right) \\cdot \\frac{4}{9}} = \\mathrm{e}^{-\\frac{12}{5} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{215}{66}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{3}{2}}}{\\sqrt[3]{\\mathrm{e}^{2\\cdot x}}}\\right)^{-\\frac{11}{9}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{215}{66}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{3}{2}}}{\\sqrt[3]{\\mathrm{e}^{2\\cdot x}}}\\right)^{-\\frac{11}{9}} = \\left(\\frac{\\mathrm{e}^{\\frac{215}{66}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{3}{2}\\cdot x}}{\\mathrm{e}^{\\frac{2}{3}\\cdot x}}\\right)^{-\\frac{11}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(\\frac{215}{66} + \\left(-\\frac{3}{2}\\right) - \\frac{2}{3}\\right)} \\right)^{-\\frac{11}{9}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(\\frac{215}{66} + \\left(-\\frac{99}{66}\\right) - \\frac{44}{66}\\right)} \\right)^{-\\frac{11}{9}} = \\left(\\mathrm{e}^{x \\cdot \\frac{72}{66}} \\right)^{-\\frac{11}{9}} = \\left(\\mathrm{e}^{x \\cdot \\frac{12}{11}} \\right)^{-\\frac{11}{9}} = $\n$\\mathrm{e}^{x \\cdot \\frac{12}{11} \\cdot \\left(-\\frac{11}{9}\\right)} = \\mathrm{e}^{-\\frac{4}{3} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{103}{36}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{9}}}{\\sqrt[6]{\\mathrm{e}^{-11\\cdot x}}}\\right)^{\\frac{1}{2}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{-\\frac{103}{36}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{2}{9}}}{\\sqrt[6]{\\mathrm{e}^{-11\\cdot x}}}\\right)^{\\frac{1}{2}} = \\left(\\frac{\\mathrm{e}^{-\\frac{103}{36}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{2}{9}\\cdot x}}{\\mathrm{e}^{-\\frac{11}{6}\\cdot x}}\\right)^{\\frac{1}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{103}{36} + \\left(-\\frac{2}{9}\\right) - \\left(-\\frac{11}{6}\\right)\\right)} \\right)^{\\frac{1}{2}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{103}{36} + \\left(-\\frac{8}{36}\\right) - \\left(-\\frac{66}{36}\\right)\\right)} \\right)^{\\frac{1}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{45}{36}\\right)} \\right)^{\\frac{1}{2}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{5}{4}\\right)} \\right)^{\\frac{1}{2}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{5}{4}\\right) \\cdot \\frac{1}{2}} = \\mathrm{e}^{-\\frac{5}{8} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{103}{54}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{1}{6}}}{\\sqrt[3]{\\mathrm{e}^{4\\cdot x}}}\\right)^{-\\frac{3}{4}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{103}{54}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{\\frac{1}{6}}}{\\sqrt[3]{\\mathrm{e}^{4\\cdot x}}}\\right)^{-\\frac{3}{4}} = \\left(\\frac{\\mathrm{e}^{\\frac{103}{54}\\cdot x} \\cdot \\mathrm{e}^{\\frac{1}{6}\\cdot x}}{\\mathrm{e}^{\\frac{4}{3}\\cdot x}}\\right)^{-\\frac{3}{4}} = \\left(\\mathrm{e}^{x \\cdot \\left(\\frac{103}{54} + \\frac{1}{6} - \\frac{4}{3}\\right)} \\right)^{-\\frac{3}{4}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(\\frac{103}{54} + \\frac{9}{54} - \\frac{72}{54}\\right)} \\right)^{-\\frac{3}{4}} = \\left(\\mathrm{e}^{x \\cdot \\frac{40}{54}} \\right)^{-\\frac{3}{4}} = \\left(\\mathrm{e}^{x \\cdot \\frac{20}{27}} \\right)^{-\\frac{3}{4}} = $\n$\\mathrm{e}^{x \\cdot \\frac{20}{27} \\cdot \\left(-\\frac{3}{4}\\right)} = \\mathrm{e}^{-\\frac{5}{9} \\cdot x}$"], ["$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{45}{13}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{9}{4}}}{\\sqrt[4]{\\mathrm{e}^{9\\cdot x}}}\\right)^{\\frac{4}{9}}$", "$\\displaystyle \\left(\\frac{\\mathrm{e}^{\\frac{45}{13}\\cdot x} \\cdot \\left(\\mathrm{e}^x\\right)^{-\\frac{9}{4}}}{\\sqrt[4]{\\mathrm{e}^{9\\cdot x}}}\\right)^{\\frac{4}{9}} = \\left(\\frac{\\mathrm{e}^{\\frac{45}{13}\\cdot x} \\cdot \\mathrm{e}^{-\\frac{9}{4}\\cdot x}}{\\mathrm{e}^{\\frac{9}{4}\\cdot x}}\\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(\\frac{45}{13} + \\left(-\\frac{9}{4}\\right) - \\frac{9}{4}\\right)} \\right)^{\\frac{4}{9}} = $\n$\\left(\\mathrm{e}^{x \\cdot \\left(\\frac{180}{52} + \\left(-\\frac{117}{52}\\right) - \\frac{117}{52}\\right)} \\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{54}{52}\\right)} \\right)^{\\frac{4}{9}} = \\left(\\mathrm{e}^{x \\cdot \\left(-\\frac{27}{26}\\right)} \\right)^{\\frac{4}{9}} = $\n$\\mathrm{e}^{x \\cdot \\left(-\\frac{27}{26}\\right) \\cdot \\frac{4}{9}} = \\mathrm{e}^{-\\frac{6}{13} \\cdot x}$"]], "     ");

=== Freitag 9. März 2018 === Schreiben Sie als Linearkombination (Summe von Vielfachen) von $\log_a(x)$ und $\log_a(y)$ miniAufgabe("#exologlaws","#solloglaws", [["$\\log_a\\left(\\left(x^{-2}y^{-3}\\right)^{4}\\right)+\\log_a\\left(\\left(x^{-5}y^{-3}\\right)^{5}\\right)$", "$-33 \\log_a(x) -27\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{5}y^{5}\\right)^{3}\\right)+\\log_a\\left(\\left(x^{-2}y^{-3}\\right)^{-5}\\right)$", "$25 \\log_a(x) +30\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{5}y^{2}\\right)^{-3}\\right)+\\log_a\\left(\\left(x^{-5}y^{-4}\\right)^{-3}\\right)$", "$0 \\log_a(x) +6\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{5}y^{-2}\\right)^{-2}\\right)+\\log_a\\left(\\left(x^{-2}y^{-2}\\right)^{5}\\right)$", "$-20 \\log_a(x) -6\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{4}y^{-5}\\right)^{3}\\right)+\\log_a\\left(\\left(x^{-2}y^{-2}\\right)^{4}\\right)$", "$4 \\log_a(x) -23\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{-3}y^{2}\\right)^{3}\\right)+\\log_a\\left(\\left(x^{-5}y^{-5}\\right)^{-5}\\right)$", "$16 \\log_a(x) +31\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{4}y^{5}\\right)^{2}\\right)+\\log_a\\left(\\left(x^{4}y^{-5}\\right)^{2}\\right)$", "$16 \\log_a(x) +0\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{4}y^{5}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{5}y^{-4}\\right)^{-3}\\right)$", "$5 \\log_a(x) +37\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{2}y^{2}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{3}y^{-4}\\right)^{2}\\right)$", "$16 \\log_a(x) +2\\log_a(y)$"], ["$\\log_a\\left(\\left(x^{2}y^{5}\\right)^{-4}\\right)+\\log_a\\left(\\left(x^{2}y^{-4}\\right)^{-3}\\right)$", "$-14 \\log_a(x) -8\\log_a(y)$"]], "
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