<PRELOAD>
miniaufgabe.js
</PRELOAD>

==== 23. April 2018 bis 27. April 2018 ====

=== Dienstag 24. April 2018 ===
Klammern Sie die jeweils kleinste Potenz von $x$ und $y$ aus, sowie ein Bruch so, dass in der Klammer kleinstmögliche ganze Zahlen als Koeffizienten vorkommen.
<JS>miniAufgabe("#exoausklammern","#solausklammern",
[["$\\displaystyle \\frac{2}{5} \\cdot x^{6} y^{3} +\\frac{8}{9} \\cdot x^{8}y^{-3}$", "$\\displaystyle \\frac{2}{45}\\left(9 \\cdot x^{6} y^{3} +20 \\cdot x^{8}y^{-3}\\right)=\\frac{2}{45}x^{6}y^{-3}\\left(9y^{6}+20x^{2}\\right)$"], ["$\\displaystyle \\frac{8}{7} \\cdot x^{-6} y^{4} +\\frac{6}{5} \\cdot x^{10}y^{3}$", "$\\displaystyle \\frac{2}{35}\\left(20 \\cdot x^{-6} y^{4} +21 \\cdot x^{10}y^{3}\\right)=\\frac{2}{35}x^{-6}y^{3}\\left(20y^{1}+21x^{16}\\right)$"], ["$\\displaystyle \\frac{3}{5} \\cdot x^{-9} y^{5} +\\frac{9}{8} \\cdot x^{17}y^{8}$", "$\\displaystyle \\frac{3}{40}\\left(8 \\cdot x^{-9} y^{5} +15 \\cdot x^{17}y^{8}\\right)=\\frac{3}{40}x^{-9}y^{5}\\left(8+15x^{26}y^{3}\\right)$"], ["$\\displaystyle \\frac{6}{5} \\cdot x^{-9} y^{-3} +\\frac{2}{7} \\cdot x^{14}y^{2}$", "$\\displaystyle \\frac{2}{35}\\left(21 \\cdot x^{-9} y^{-3} +5 \\cdot x^{14}y^{2}\\right)=\\frac{2}{35}x^{-9}y^{-3}\\left(21+5x^{23}y^{5}\\right)$"], ["$\\displaystyle -\\frac{2}{9} \\cdot x^{6} y^{4} +\\frac{6}{7} \\cdot x^{11}y^{-5}$", "$\\displaystyle \\frac{2}{63}\\left(-7 \\cdot x^{6} y^{4} +27 \\cdot x^{11}y^{-5}\\right)=\\frac{2}{63}x^{6}y^{-5}\\left(-7y^{9}+27x^{5}\\right)$"], ["$\\displaystyle \\frac{2}{3} \\cdot x^{-4} y^{3} -\\frac{4}{5} \\cdot x^{-6}y^{-3}$", "$\\displaystyle \\frac{2}{15}\\left(5 \\cdot x^{-4} y^{3} -6 \\cdot x^{-6}y^{-3}\\right)=\\frac{2}{15}x^{-6}y^{-3}\\left(5x^{2}y^{6}-6\\right)$"], ["$\\displaystyle -\\frac{8}{3} \\cdot x^{3} y^{-4} -\\frac{6}{7} \\cdot x^{-11}y^{8}$", "$\\displaystyle \\frac{2}{21}\\left(-28 \\cdot x^{3} y^{-4} -9 \\cdot x^{-11}y^{8}\\right)=\\frac{2}{21}x^{-11}y^{-4}\\left(-28x^{14}-9y^{12}\\right)$"], ["$\\displaystyle -\\frac{3}{4} \\cdot x^{4} y^{-6} -\\frac{9}{5} \\cdot x^{8}y^{4}$", "$\\displaystyle \\frac{3}{20}\\left(-5 \\cdot x^{4} y^{-6} -12 \\cdot x^{8}y^{4}\\right)=\\frac{3}{20}x^{4}y^{-6}\\left(-5-12x^{4}y^{10}\\right)$"], ["$\\displaystyle \\frac{6}{5} \\cdot x^{-2} y^{7} +\\frac{2}{7} \\cdot x^{-16}y^{-3}$", "$\\displaystyle \\frac{2}{35}\\left(21 \\cdot x^{-2} y^{7} +5 \\cdot x^{-16}y^{-3}\\right)=\\frac{2}{35}x^{-16}y^{-3}\\left(21x^{14}y^{10}+5\\right)$"], ["$\\displaystyle \\frac{8}{9} \\cdot x^{-9} y^{7} -\\frac{8}{7} \\cdot x^{18}y^{-9}$", "$\\displaystyle \\frac{8}{63}\\left(7 \\cdot x^{-9} y^{7} -9 \\cdot x^{18}y^{-9}\\right)=\\frac{8}{63}x^{-9}y^{-9}\\left(7y^{16}-9x^{27}\\right)$"]],
" <br><hr> ", " <br><hr> ", 3);
</JS>
<HTML>
<div id="exoausklammern"></div>

</HTML>
<hidden Lösungen>
<HTML>
<div id="solausklammern"></div>
</HTML>
</hidden>

=== Freitag 27. April 2018 ===
Vereinfachen Sie:
<JS>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}$"]],
" &nbsp; &nbsp; ");
</JS>
<HTML>
<div id="exoehochxbrueche"></div>

</HTML>
<hidden Lösungen>
<HTML>
<div id="solehochxbrueche"></div>
</HTML>
</hidden>