<PRELOAD>
miniaufgabe.js
</PRELOAD>
=== Dienstag 20. August 2019 ===
Leiten Sie von Hand und ohne Unterlagen ab:
<JS>miniAufgabe("#exonurpolynome","#solnurpolynome",
[["a) $f(x)=\\frac{1}{6}x^{12}+\\frac{4}{7}x^{9}+\\frac{1}{4}x^{7}\\quad$ b) $f(x)=-\\frac{2}{3}x^{9}-\\frac{2}{3}x^{3}+\\frac{4}{9}x^{2}\\quad$ ", "a) $f'(x)=2x^{11}+\\frac{36}{7}x^{8}+\\frac{7}{4}x^{6}\\quad$ b) $f'(x)=-6x^{8}-2x^{2}+\\frac{8}{9}x\\quad$ "], ["a) $f(x)=\\frac{1}{2}x^{12}+\\frac{1}{3}x^{11}-\\frac{1}{7}x^{5}\\quad$ b) $f(x)=-\\frac{1}{4}x^{9}+\\frac{4}{7}x^{4}+\\frac{4}{9}x^{3}\\quad$ ", "a) $f'(x)=6x^{11}+\\frac{11}{3}x^{10}-\\frac{5}{7}x^{4}\\quad$ b) $f'(x)=-\\frac{9}{4}x^{8}+\\frac{16}{7}x^{3}+\\frac{4}{3}x^{2}\\quad$ "], ["a) $f(x)=-\\frac{2}{7}x^{12}+\\frac{3}{8}x^{7}-\\frac{1}{4}x^{4}\\quad$ b) $f(x)=-\\frac{4}{9}x^{11}-\\frac{1}{3}x^{7}+\\frac{2}{5}x^{3}\\quad$ ", "a) $f'(x)=-\\frac{24}{7}x^{11}+\\frac{21}{8}x^{6}-x^{3}\\quad$ b) $f'(x)=-\\frac{44}{9}x^{10}-\\frac{7}{3}x^{6}+\\frac{6}{5}x^{2}\\quad$ "], ["a) $f(x)=-\\frac{2}{7}x^{11}+\\frac{1}{5}x^{9}+\\frac{3}{7}x^{4}\\quad$ b) $f(x)=-\\frac{3}{4}x^{11}+\\frac{1}{5}x^{6}-\\frac{2}{9}x^{2}\\quad$ ", "a) $f'(x)=-\\frac{22}{7}x^{10}+\\frac{9}{5}x^{8}+\\frac{12}{7}x^{3}\\quad$ b) $f'(x)=-\\frac{33}{4}x^{10}+\\frac{6}{5}x^{5}-\\frac{4}{9}x\\quad$ "], ["a) $f(x)=\\frac{1}{3}x^{10}-\\frac{1}{3}x^{8}+\\frac{3}{4}x^{4}\\quad$ b) $f(x)=-\\frac{1}{6}x^{12}-\\frac{1}{3}x^{8}+\\frac{1}{2}x^{2}\\quad$ ", "a) $f'(x)=\\frac{10}{3}x^{9}-\\frac{8}{3}x^{7}+3x^{3}\\quad$ b) $f'(x)=-2x^{11}-\\frac{8}{3}x^{7}+x\\quad$ "], ["a) $f(x)=\\frac{1}{8}x^{11}+\\frac{1}{4}x^{5}+\\frac{1}{2}x^{4}\\quad$ b) $f(x)=\\frac{2}{3}x^{12}-\\frac{4}{9}x^{4}+\\frac{4}{7}x^{2}\\quad$ ", "a) $f'(x)=\\frac{11}{8}x^{10}+\\frac{5}{4}x^{4}+2x^{3}\\quad$ b) $f'(x)=8x^{11}-\\frac{16}{9}x^{3}+\\frac{8}{7}x\\quad$ "], ["a) $f(x)=\\frac{4}{3}x^{11}-\\frac{4}{5}x^{9}-\\frac{1}{4}x^{7}\\quad$ b) $f(x)=-\\frac{1}{3}x^{12}+\\frac{2}{3}x^{9}-\\frac{1}{5}x^{6}\\quad$ ", "a) $f'(x)=\\frac{44}{3}x^{10}-\\frac{36}{5}x^{8}-\\frac{7}{4}x^{6}\\quad$ b) $f'(x)=-4x^{11}+6x^{8}-\\frac{6}{5}x^{5}\\quad$ "], ["a) $f(x)=\\frac{1}{6}x^{5}+\\frac{1}{3}x^{3}+\\frac{4}{9}x^{2}\\quad$ b) $f(x)=\\frac{3}{4}x^{7}+\\frac{1}{2}x^{3}-\\frac{1}{6}x^{2}\\quad$ ", "a) $f'(x)=\\frac{5}{6}x^{4}+x^{2}+\\frac{8}{9}x\\quad$ b) $f'(x)=\\frac{21}{4}x^{6}+\\frac{3}{2}x^{2}-\\frac{1}{3}x\\quad$ "], ["a) $f(x)=\\frac{1}{8}x^{10}-\\frac{2}{3}x^{7}-\\frac{3}{8}x^{3}\\quad$ b) $f(x)=\\frac{1}{2}x^{8}+\\frac{1}{3}x^{4}-\\frac{2}{3}x^{3}\\quad$ ", "a) $f'(x)=\\frac{5}{4}x^{9}-\\frac{14}{3}x^{6}-\\frac{9}{8}x^{2}\\quad$ b) $f'(x)=4x^{7}+\\frac{4}{3}x^{3}-2x^{2}\\quad$ "], ["a) $f(x)=-\\frac{2}{7}x^{9}-\\frac{1}{3}x^{7}+\\frac{1}{9}x^{4}\\quad$ b) $f(x)=-\\frac{1}{9}x^{12}-\\frac{1}{2}x^{4}-\\frac{1}{3}x^{2}\\quad$ ", "a) $f'(x)=-\\frac{18}{7}x^{8}-\\frac{7}{3}x^{6}+\\frac{4}{9}x^{3}\\quad$ b) $f'(x)=-\\frac{4}{3}x^{11}-2x^{3}-\\frac{2}{3}x\\quad$ "]],
" <br> ");
</JS>
<HTML>
<div id="exonurpolynome"></div>

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


=== Donnerstag 22. August 2019 ===

Nötige Potenzgesetze: 
  * $\left(a^b\right)^c = a^{b \cdot c}$
  * $(a\cdot b)^c = a^c \cdot b^c$
  * $a^{-n} = \frac{1}{a^n}$
  * $\frac{a^n}{a^m} = a^{n-m} = \frac{1}{a^{m-n}}$

Vereinfachen Sie soweit wie möglich. Schreiben Sie das Resultat als Bruch aus Produkten aus Potenzen mit positiven Exponenten.
  - $\left(\frac{\left(c^{-2} \cdot b^{-3}\right)^{-3}}{\left(a^{-4} \cdot b^{2}\right)^{5}}\right)^{2}$
  - $\left(\frac{\left(c^{5} \cdot a^{-2}\right)^{2}}{\left(b^{-1} \cdot c^{3}\right)^{-2}}\right)^{-3}$
  - $\left(\frac{\left(a^{2} \cdot c^{4}\right)^{-3}}{\left(c^{-1} \cdot b^{-2}\right)^{-4}}\right)^{3}$
<hidden Lösungsvorschlag>
  - $\left(\frac{\left(c^{-2} \cdot b^{-3}\right)^{-3}}{\left(a^{-4} \cdot b^{2}\right)^{5}}\right)^{2} = \left(\frac{c^{6} \cdot b^{9}}{a^{-20} \cdot b^{10}}\right)^{2} = \left(\frac{c^{6}}{a^{-20} \cdot b}\right)^{2} = \frac{c^{12}}{a^{-40} \cdot b^2} = \frac{c^{12} \cdot a^{40}}{b^2}$
  - $\left(\frac{\left(c^{5} \cdot a^{-2}\right)^{2}}{\left(b^{-1} \cdot c^{3}\right)^{-2}}\right)^{-3} = \left(\frac{c^{10} \cdot a^{-4}}{b^{2} \cdot c^{-6}}\right)^{-3} = \left(\frac{c^{16} \cdot a^{-4}}{b^{2}}\right)^{-3} = \left(\frac{b^{2}}{c^{16} \cdot a^{-4}}\right)^{3} = \frac{b^{6}}{c^{48} \cdot a^{-12}} = \frac{b^{6} \cdot a^{12}}{c^{48}}$
  - $\left(\frac{\left(a^{2} \cdot c^{4}\right)^{-3}}{\left(c^{-1} \cdot b^{-2}\right)^{-4}}\right)^{3} = \left(\frac{a^{-6} \cdot c^{-12}}{c^{4} \cdot b^{8}}\right)^{3} = \left(\frac{a^{-6}}{c^{16} \cdot b^{8}}\right)^{3} =\frac{a^{-18}}{c^{48} \cdot b^{24}} = \frac{1}{a^{18}b^{24}c^{48}}$
</hidden>