miniaufgabe.js ==== 30. Mai 2022 bis 3. Juni 2022 ==== === Donnerstag 2. Juni 2022 === Resultat als Bruch von Produkten von Potenzen mit positiven Exponenten.miniAufgabe("#exonegex_varsimple","#solnegex_varsimple", [["$\\displaystyle \\left(a^{-6}b^{-5}h^{2}\\right)^{6}$", "$\\displaystyle \\left(a^{-6}b^{-5}h^{2}\\right)^{6} = a^{-36}b^{-30}h^{12} = \\frac{h^{12}}{a^{36}b^{30}}$"], ["$\\displaystyle \\left(a^{-4}f^{-3}y^{2}\\right)^{4}$", "$\\displaystyle \\left(a^{-4}f^{-3}y^{2}\\right)^{4} = a^{-16}f^{-12}y^{8} = \\frac{y^{8}}{a^{16}f^{12}}$"], ["$\\displaystyle \\left(b^{4}x^{-5}y^{-6}\\right)^{-5}$", "$\\displaystyle \\left(b^{4}x^{-5}y^{-6}\\right)^{-5} = b^{-20}x^{25}y^{30} = \\frac{x^{25}y^{30}}{b^{20}}$"], ["$\\displaystyle \\left(e^{5}h^{-3}y^{-4}\\right)^{-3}$", "$\\displaystyle \\left(e^{5}h^{-3}y^{-4}\\right)^{-3} = e^{-15}h^{9}y^{12} = \\frac{h^{9}y^{12}}{e^{15}}$"], ["$\\displaystyle \\left(e^{-5}m^{2}y^{-3}\\right)^{5}$", "$\\displaystyle \\left(e^{-5}m^{2}y^{-3}\\right)^{5} = e^{-25}m^{10}y^{-15} = \\frac{m^{10}}{e^{25}y^{15}}$"], ["$\\displaystyle \\left(f^{-3}h^{2}y^{-5}\\right)^{-5}$", "$\\displaystyle \\left(f^{-3}h^{2}y^{-5}\\right)^{-5} = f^{15}h^{-10}y^{25} = \\frac{f^{15}y^{25}}{h^{10}}$"], ["$\\displaystyle \\left(d^{-3}u^{-2}x^{4}\\right)^{6}$", "$\\displaystyle \\left(d^{-3}u^{-2}x^{4}\\right)^{6} = d^{-18}u^{-12}x^{24} = \\frac{x^{24}}{d^{18}u^{12}}$"], ["$\\displaystyle \\left(e^{-6}p^{3}x^{-2}\\right)^{2}$", "$\\displaystyle \\left(e^{-6}p^{3}x^{-2}\\right)^{2} = e^{-12}p^{6}x^{-4} = \\frac{p^{6}}{e^{12}x^{4}}$"], ["$\\displaystyle \\left(c^{-6}d^{5}e^{-2}\\right)^{6}$", "$\\displaystyle \\left(c^{-6}d^{5}e^{-2}\\right)^{6} = c^{-36}d^{30}e^{-12} = \\frac{d^{30}}{c^{36}e^{12}}$"], ["$\\displaystyle \\left(f^{-3}k^{4}n^{-2}\\right)^{5}$", "$\\displaystyle \\left(f^{-3}k^{4}n^{-2}\\right)^{5} = f^{-15}k^{20}n^{-10} = \\frac{k^{20}}{f^{15}n^{10}}$"]], "
", "
");
ruby potenzen-und-brueche.rb 12
 === Freitag 3. Juni 2022 === Resultat als gekürzten Bruch von Produkten von Potenzen mit positiven Exponenten.miniAufgabe("#exopotenzenBruecheVariablen1","#solpotenzenBruecheVariablen1", [["$\\displaystyle \\left(\\frac{\\left(m^{2} \\cdot d^{3}\\right)^{5}}{\\left(d^{-5} \\cdot c^{-2}\\right)^{-4}}\\right)^{-5}$", "$\\left(\\frac{\\left(m^{2} \\cdot d^{3}\\right)^{5}}{\\left(d^{-5} \\cdot c^{-2}\\right)^{-4}}\\right)^{-5} = \\left(\\frac{\\left(m^{2}\\right)^{5} \\cdot \\left(d^{3}\\right)^{5}}{\\left(d^{-5}\\right)^{-4} \\cdot \\left(c^{-2}\\right)^{-4}}\\right)^{-5} = \\left(\\frac{m^{2 \\cdot 5} \\cdot d^{3 \\cdot 5}}{d^{-5 \\cdot -4} \\cdot c^{-2 \\cdot -4}}\\right)^{-5} = \\left(\\frac{m^{10} \\cdot d^{15}}{d^{20} \\cdot c^{8}}\\right)^{-5} = \\left(\\frac{m^{10}}{c^{8}d^{5}}\\right)^{-5} = \\frac{\\left(m^{10}\\right)^{-5}}{\\left(c^{8}d^{5}\\right)^{-5}} = \\frac{m^{10 \\cdot -5}}{\\left(c^{8}\\right)^{-5} \\cdot \\left(d^{5}\\right)^{-5}} = \\frac{m^{-50}}{c^{8 \\cdot -5} \\cdot d^{5 \\cdot -5}} = \\frac{m^{-50}}{c^{-40} \\cdot d^{-25}} = m^{-50}c^{40}d^{25} = c^{40}d^{25}m^{-50} = \\frac{c^{40}d^{25}}{m^{50}}$"], ["$\\displaystyle \\left(\\frac{\\left(w^{5} \\cdot f^{-4}\\right)^{-3}}{\\left(f^{-3} \\cdot a^{3}\\right)^{3}}\\right)^{4}$", "$\\left(\\frac{\\left(w^{5} \\cdot f^{-4}\\right)^{-3}}{\\left(f^{-3} \\cdot a^{3}\\right)^{3}}\\right)^{4} = \\left(\\frac{\\left(w^{5}\\right)^{-3} \\cdot \\left(f^{-4}\\right)^{-3}}{\\left(f^{-3}\\right)^{3} \\cdot \\left(a^{3}\\right)^{3}}\\right)^{4} = \\left(\\frac{w^{5 \\cdot -3} \\cdot f^{-4 \\cdot -3}}{f^{-3 \\cdot 3} \\cdot a^{3 \\cdot 3}}\\right)^{4} = \\left(\\frac{w^{-15} \\cdot f^{12}}{f^{-9} \\cdot a^{9}}\\right)^{4} = \\left(\\frac{f^{21}w^{-15}}{a^{9}}\\right)^{4} = \\frac{\\left(f^{21}w^{-15}\\right)^{4}}{\\left(a^{9}\\right)^{4}} = \\frac{\\left(f^{21}\\right)^{4} \\cdot \\left(w^{-15}\\right)^{4}}{a^{9 \\cdot 4}} = \\frac{f^{21 \\cdot 4} \\cdot w^{-15 \\cdot 4}}{a^{36}} = \\frac{f^{84} \\cdot w^{-60}}{a^{36}} = f^{84}w^{-60}a^{-36} = a^{-36}f^{84}w^{-60} = \\frac{f^{84}}{a^{36}w^{60}}$"], ["$\\displaystyle \\left(\\frac{\\left(n^{-4} \\cdot k^{-2}\\right)^{2}}{\\left(k^{3} \\cdot u^{-4}\\right)^{2}}\\right)^{3}$", "$\\left(\\frac{\\left(n^{-4} \\cdot k^{-2}\\right)^{2}}{\\left(k^{3} \\cdot u^{-4}\\right)^{2}}\\right)^{3} = \\left(\\frac{\\left(n^{-4}\\right)^{2} \\cdot \\left(k^{-2}\\right)^{2}}{\\left(k^{3}\\right)^{2} \\cdot \\left(u^{-4}\\right)^{2}}\\right)^{3} = \\left(\\frac{n^{-4 \\cdot 2} \\cdot k^{-2 \\cdot 2}}{k^{3 \\cdot 2} \\cdot u^{-4 \\cdot 2}}\\right)^{3} = \\left(\\frac{n^{-8} \\cdot k^{-4}}{k^{6} \\cdot u^{-8}}\\right)^{3} = \\left(\\frac{n^{-8}}{k^{10}u^{-8}}\\right)^{3} = \\frac{\\left(n^{-8}\\right)^{3}}{\\left(k^{10}u^{-8}\\right)^{3}} = \\frac{n^{-8 \\cdot 3}}{\\left(k^{10}\\right)^{3} \\cdot \\left(u^{-8}\\right)^{3}} = \\frac{n^{-24}}{k^{10 \\cdot 3} \\cdot u^{-8 \\cdot 3}} = \\frac{n^{-24}}{k^{30} \\cdot u^{-24}} = n^{-24}k^{-30}u^{24} = k^{-30}n^{-24}u^{24} = \\frac{u^{24}}{k^{30}n^{24}}$"], ["$\\displaystyle \\left(\\frac{\\left(d^{-5} \\cdot f^{3}\\right)^{-3}}{\\left(f^{-3} \\cdot x^{-2}\\right)^{-4}}\\right)^{5}$", "$\\left(\\frac{\\left(d^{-5} \\cdot f^{3}\\right)^{-3}}{\\left(f^{-3} \\cdot x^{-2}\\right)^{-4}}\\right)^{5} = \\left(\\frac{\\left(d^{-5}\\right)^{-3} \\cdot \\left(f^{3}\\right)^{-3}}{\\left(f^{-3}\\right)^{-4} \\cdot \\left(x^{-2}\\right)^{-4}}\\right)^{5} = \\left(\\frac{d^{-5 \\cdot -3} \\cdot f^{3 \\cdot -3}}{f^{-3 \\cdot -4} \\cdot x^{-2 \\cdot -4}}\\right)^{5} = \\left(\\frac{d^{15} \\cdot f^{-9}}{f^{12} \\cdot x^{8}}\\right)^{5} = \\left(\\frac{d^{15}}{f^{21}x^{8}}\\right)^{5} = \\frac{\\left(d^{15}\\right)^{5}}{\\left(f^{21}x^{8}\\right)^{5}} = \\frac{d^{15 \\cdot 5}}{\\left(f^{21}\\right)^{5} \\cdot \\left(x^{8}\\right)^{5}} = \\frac{d^{75}}{f^{21 \\cdot 5} \\cdot x^{8 \\cdot 5}} = \\frac{d^{75}}{f^{105} \\cdot x^{40}} = d^{75}f^{-105}x^{-40} = \\frac{d^{75}}{f^{105}x^{40}}$"], ["$\\displaystyle \\left(\\frac{\\left(y^{5} \\cdot p^{3}\\right)^{4}}{\\left(p^{4} \\cdot a^{2}\\right)^{-5}}\\right)^{-3}$", "$\\left(\\frac{\\left(y^{5} \\cdot p^{3}\\right)^{4}}{\\left(p^{4} \\cdot a^{2}\\right)^{-5}}\\right)^{-3} = \\left(\\frac{\\left(y^{5}\\right)^{4} \\cdot \\left(p^{3}\\right)^{4}}{\\left(p^{4}\\right)^{-5} \\cdot \\left(a^{2}\\right)^{-5}}\\right)^{-3} = \\left(\\frac{y^{5 \\cdot 4} \\cdot p^{3 \\cdot 4}}{p^{4 \\cdot -5} \\cdot a^{2 \\cdot -5}}\\right)^{-3} = \\left(\\frac{y^{20} \\cdot p^{12}}{p^{-20} \\cdot a^{-10}}\\right)^{-3} = \\left(\\frac{p^{32}y^{20}}{a^{-10}}\\right)^{-3} = \\frac{\\left(p^{32}y^{20}\\right)^{-3}}{\\left(a^{-10}\\right)^{-3}} = \\frac{\\left(p^{32}\\right)^{-3} \\cdot \\left(y^{20}\\right)^{-3}}{a^{-10 \\cdot -3}} = \\frac{p^{32 \\cdot -3} \\cdot y^{20 \\cdot -3}}{a^{30}} = \\frac{p^{-96} \\cdot y^{-60}}{a^{30}} = p^{-96}y^{-60}a^{-30} = a^{-30}p^{-96}y^{-60} = \\frac{}{a^{30}p^{96}y^{60}}$"], ["$\\displaystyle \\left(\\frac{\\left(y^{4} \\cdot d^{2}\\right)^{4}}{\\left(d^{4} \\cdot p^{-3}\\right)^{-2}}\\right)^{-3}$", "$\\left(\\frac{\\left(y^{4} \\cdot d^{2}\\right)^{4}}{\\left(d^{4} \\cdot p^{-3}\\right)^{-2}}\\right)^{-3} = \\left(\\frac{\\left(y^{4}\\right)^{4} \\cdot \\left(d^{2}\\right)^{4}}{\\left(d^{4}\\right)^{-2} \\cdot \\left(p^{-3}\\right)^{-2}}\\right)^{-3} = \\left(\\frac{y^{4 \\cdot 4} \\cdot d^{2 \\cdot 4}}{d^{4 \\cdot -2} \\cdot p^{-3 \\cdot -2}}\\right)^{-3} = \\left(\\frac{y^{16} \\cdot d^{8}}{d^{-8} \\cdot p^{6}}\\right)^{-3} = \\left(\\frac{d^{16}y^{16}}{p^{6}}\\right)^{-3} = \\frac{\\left(d^{16}y^{16}\\right)^{-3}}{\\left(p^{6}\\right)^{-3}} = \\frac{\\left(d^{16}\\right)^{-3} \\cdot \\left(y^{16}\\right)^{-3}}{p^{6 \\cdot -3}} = \\frac{d^{16 \\cdot -3} \\cdot y^{16 \\cdot -3}}{p^{-18}} = \\frac{d^{-48} \\cdot y^{-48}}{p^{-18}} = d^{-48}y^{-48}p^{18} = d^{-48}p^{18}y^{-48} = \\frac{p^{18}}{d^{48}y^{48}}$"], ["$\\displaystyle \\left(\\frac{\\left(e^{3} \\cdot k^{3}\\right)^{-2}}{\\left(k^{-2} \\cdot p^{2}\\right)^{3}}\\right)^{-4}$", "$\\left(\\frac{\\left(e^{3} \\cdot k^{3}\\right)^{-2}}{\\left(k^{-2} \\cdot p^{2}\\right)^{3}}\\right)^{-4} = \\left(\\frac{\\left(e^{3}\\right)^{-2} \\cdot \\left(k^{3}\\right)^{-2}}{\\left(k^{-2}\\right)^{3} \\cdot \\left(p^{2}\\right)^{3}}\\right)^{-4} = \\left(\\frac{e^{3 \\cdot -2} \\cdot k^{3 \\cdot -2}}{k^{-2 \\cdot 3} \\cdot p^{2 \\cdot 3}}\\right)^{-4} = \\left(\\frac{e^{-6} \\cdot k^{-6}}{k^{-6} \\cdot p^{6}}\\right)^{-4} = \\left(\\frac{e^{-6}}{p^{6}}\\right)^{-4} = \\frac{\\left(e^{-6}\\right)^{-4}}{\\left(p^{6}\\right)^{-4}} = \\frac{e^{-6 \\cdot -4}}{p^{6 \\cdot -4}} = \\frac{e^{24}}{p^{-24}} = e^{24}p^{24}$"], ["$\\displaystyle \\left(\\frac{\\left(e^{2} \\cdot b^{-4}\\right)^{2}}{\\left(b^{-2} \\cdot d^{4}\\right)^{-2}}\\right)^{2}$", "$\\left(\\frac{\\left(e^{2} \\cdot b^{-4}\\right)^{2}}{\\left(b^{-2} \\cdot d^{4}\\right)^{-2}}\\right)^{2} = \\left(\\frac{\\left(e^{2}\\right)^{2} \\cdot \\left(b^{-4}\\right)^{2}}{\\left(b^{-2}\\right)^{-2} \\cdot \\left(d^{4}\\right)^{-2}}\\right)^{2} = \\left(\\frac{e^{2 \\cdot 2} \\cdot b^{-4 \\cdot 2}}{b^{-2 \\cdot -2} \\cdot d^{4 \\cdot -2}}\\right)^{2} = \\left(\\frac{e^{4} \\cdot b^{-8}}{b^{4} \\cdot d^{-8}}\\right)^{2} = \\left(\\frac{e^{4}}{b^{12}d^{-8}}\\right)^{2} = \\frac{\\left(e^{4}\\right)^{2}}{\\left(b^{12}d^{-8}\\right)^{2}} = \\frac{e^{4 \\cdot 2}}{\\left(b^{12}\\right)^{2} \\cdot \\left(d^{-8}\\right)^{2}} = \\frac{e^{8}}{b^{12 \\cdot 2} \\cdot d^{-8 \\cdot 2}} = \\frac{e^{8}}{b^{24} \\cdot d^{-16}} = e^{8}b^{-24}d^{16} = b^{-24}d^{16}e^{8} = \\frac{d^{16}e^{8}}{b^{24}}$"], ["$\\displaystyle \\left(\\frac{\\left(x^{-4} \\cdot k^{-3}\\right)^{5}}{\\left(k^{-5} \\cdot n^{2}\\right)^{-2}}\\right)^{2}$", "$\\left(\\frac{\\left(x^{-4} \\cdot k^{-3}\\right)^{5}}{\\left(k^{-5} \\cdot n^{2}\\right)^{-2}}\\right)^{2} = \\left(\\frac{\\left(x^{-4}\\right)^{5} \\cdot \\left(k^{-3}\\right)^{5}}{\\left(k^{-5}\\right)^{-2} \\cdot \\left(n^{2}\\right)^{-2}}\\right)^{2} = \\left(\\frac{x^{-4 \\cdot 5} \\cdot k^{-3 \\cdot 5}}{k^{-5 \\cdot -2} \\cdot n^{2 \\cdot -2}}\\right)^{2} = \\left(\\frac{x^{-20} \\cdot k^{-15}}{k^{10} \\cdot n^{-4}}\\right)^{2} = \\left(\\frac{x^{-20}}{k^{25}n^{-4}}\\right)^{2} = \\frac{\\left(x^{-20}\\right)^{2}}{\\left(k^{25}n^{-4}\\right)^{2}} = \\frac{x^{-20 \\cdot 2}}{\\left(k^{25}\\right)^{2} \\cdot \\left(n^{-4}\\right)^{2}} = \\frac{x^{-40}}{k^{25 \\cdot 2} \\cdot n^{-4 \\cdot 2}} = \\frac{x^{-40}}{k^{50} \\cdot n^{-8}} = x^{-40}k^{-50}n^{8} = k^{-50}n^{8}x^{-40} = \\frac{n^{8}}{k^{50}x^{40}}$"], ["$\\displaystyle \\left(\\frac{\\left(n^{-4} \\cdot y^{-4}\\right)^{-5}}{\\left(y^{4} \\cdot p^{-2}\\right)^{4}}\\right)^{4}$", "$\\left(\\frac{\\left(n^{-4} \\cdot y^{-4}\\right)^{-5}}{\\left(y^{4} \\cdot p^{-2}\\right)^{4}}\\right)^{4} = \\left(\\frac{\\left(n^{-4}\\right)^{-5} \\cdot \\left(y^{-4}\\right)^{-5}}{\\left(y^{4}\\right)^{4} \\cdot \\left(p^{-2}\\right)^{4}}\\right)^{4} = \\left(\\frac{n^{-4 \\cdot -5} \\cdot y^{-4 \\cdot -5}}{y^{4 \\cdot 4} \\cdot p^{-2 \\cdot 4}}\\right)^{4} = \\left(\\frac{n^{20} \\cdot y^{20}}{y^{16} \\cdot p^{-8}}\\right)^{4} = \\left(\\frac{n^{20}y^{4}}{p^{-8}}\\right)^{4} = \\frac{\\left(n^{20}y^{4}\\right)^{4}}{\\left(p^{-8}\\right)^{4}} = \\frac{\\left(n^{20}\\right)^{4} \\cdot \\left(y^{4}\\right)^{4}}{p^{-8 \\cdot 4}} = \\frac{n^{20 \\cdot 4} \\cdot y^{4 \\cdot 4}}{p^{-32}} = \\frac{n^{80} \\cdot y^{16}}{p^{-32}} = n^{80}y^{16}p^{32} = n^{80}p^{32}y^{16}$"]], "
", "
");
ruby potenzen-und-brueche-variablen.rb 1