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Ausführliche Lösung zu Jans Aufgabe zur Potenzrechnung (siehe Kommentar)
![\left[ \left ( \frac{-3x^{3}}{y^{4}} \right)^3\cdot\left ( \frac{4y^3}{5x^{2}} \right )^{2} \right ]:\left ( \frac{6y^{3}}{5x} \right )^{2}= \\ \\ \\ =\left [ \left ( \frac{-3^{3}x^{3\cdot 3}}{y^{4\cdot3}} \right)\cdot\left ( \frac{4^{2}y^{3\cdot 2}}{5^{2}x^{2\cdot2}}\right )\right ]:\left( \frac{6^{2}y^{3\cdot2}}{5^{2}x^{2}}\right)](http://s0.wp.com/latex.php?latex=+++%5Cleft%5B+%5Cleft+%28+%5Cfrac%7B-3x%5E%7B3%7D%7D%7By%5E%7B4%7D%7D+%5Cright%29%5E3%5Ccdot%5Cleft+%28+%5Cfrac%7B4y%5E3%7D%7B5x%5E%7B2%7D%7D+%5Cright+%29%5E%7B2%7D+%5Cright+%5D%3A%5Cleft+%28+%5Cfrac%7B6y%5E%7B3%7D%7D%7B5x%7D+%5Cright+%29%5E%7B2%7D%3D+%5C%5C+++%5C%5C+%5C%5C+++%3D%5Cleft+%5B+%5Cleft+%28+%5Cfrac%7B-3%5E%7B3%7Dx%5E%7B3%5Ccdot+3%7D%7D%7By%5E%7B4%5Ccdot3%7D%7D+%5Cright%29%5Ccdot%5Cleft+%28+%5Cfrac%7B4%5E%7B2%7Dy%5E%7B3%5Ccdot+2%7D%7D%7B5%5E%7B2%7Dx%5E%7B2%5Ccdot2%7D%7D%5Cright+%29%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B6%5E%7B2%7Dy%5E%7B3%5Ccdot2%7D%7D%7B5%5E%7B2%7Dx%5E%7B2%7D%7D%5Cright%29&bg=ffffff&fg=000000&s=2 "\left[ \left ( \frac{-3x^{3}}{y^{4}} \right)^3\cdot\left ( \frac{4y^3}{5x^{2}} \right )^{2} \right ]:\left ( \frac{6y^{3}}{5x} \right )^{2}= \\ \\ \\ =\left [ \left ( \frac{-3^{3}x^{3\cdot 3}}{y^{4\cdot3}} \right)\cdot\left ( \frac{4^{2}y^{3\cdot 2}}{5^{2}x^{2\cdot2}}\right )\right ]:\left( \frac{6^{2}y^{3\cdot2}}{5^{2}x^{2}}\right)")
Verwendete Regeln für Schritt 1
Jetzt mutiplizieren wir aus
![\left [ \left ( \frac{-3^{3}x^{3\cdot 3}}{y^{4\cdot3}} \right)\cdot\left ( \frac{4^{2}y^{3\cdot 2}}{5^{2}x^{2\cdot2}}\right )\right ]:\left( \frac{6^{2}y^{3\cdot2}}{5^{2}x^{2}}\right)=\\ \\ \\ =\left [ \left ( \frac{-27x^{9}}{y^{12}} \right)\cdot\left ( \frac{16y^{6}}{25x^{4}}\right )\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)= \\ \\ \\ =\left [ \frac{-27x^{9}16y^{6}}{y^{12}25x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)=\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)](http://s0.wp.com/latex.php?latex=+++%5Cleft+%5B+%5Cleft+%28+%5Cfrac%7B-3%5E%7B3%7Dx%5E%7B3%5Ccdot+3%7D%7D%7By%5E%7B4%5Ccdot3%7D%7D+%5Cright%29%5Ccdot%5Cleft+%28+%5Cfrac%7B4%5E%7B2%7Dy%5E%7B3%5Ccdot+2%7D%7D%7B5%5E%7B2%7Dx%5E%7B2%5Ccdot2%7D%7D%5Cright+%29%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B6%5E%7B2%7Dy%5E%7B3%5Ccdot2%7D%7D%7B5%5E%7B2%7Dx%5E%7B2%7D%7D%5Cright%29%3D%5C%5C+++%5C%5C+%5C%5C+++%3D%5Cleft+%5B+%5Cleft+%28+%5Cfrac%7B-27x%5E%7B9%7D%7D%7By%5E%7B12%7D%7D+%5Cright%29%5Ccdot%5Cleft+%28+%5Cfrac%7B16y%5E%7B6%7D%7D%7B25x%5E%7B4%7D%7D%5Cright+%29%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B36y%5E%7B6%7D%7D%7B25x%5E%7B2%7D%7D%5Cright%29%3D+%5C%5C+++%5C%5C+%5C%5C+++%3D%5Cleft+%5B+%5Cfrac%7B-27x%5E%7B9%7D16y%5E%7B6%7D%7D%7By%5E%7B12%7D25x%5E%7B4%7D%7D%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B36y%5E%7B6%7D%7D%7B25x%5E%7B2%7D%7D%5Cright%29%3D%5Cleft+%5B+%5Cfrac%7B-432x%5E%7B9%7Dy%5E%7B6%7D%7D%7B25y%5E%7B12%7Dx%5E%7B4%7D%7D%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B36y%5E%7B6%7D%7D%7B25x%5E%7B2%7D%7D%5Cright%29+++&bg=ffffff&fg=000000&s=2 "\left [ \left ( \frac{-3^{3}x^{3\cdot 3}}{y^{4\cdot3}} \right)\cdot\left ( \frac{4^{2}y^{3\cdot 2}}{5^{2}x^{2\cdot2}}\right )\right ]:\left( \frac{6^{2}y^{3\cdot2}}{5^{2}x^{2}}\right)=\\ \\ \\ =\left [ \left ( \frac{-27x^{9}}{y^{12}} \right)\cdot\left ( \frac{16y^{6}}{25x^{4}}\right )\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)= \\ \\ \\ =\left [ \frac{-27x^{9}16y^{6}}{y^{12}25x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)=\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)")
An dieser Stelle gilt die Regel „geteilt ist dasselbe wie multipliziert mit dem Kehrbruch“ (Bruchrechnung)
Bsp.
![\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)=\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ] \cdot \left( \frac{25x^{2}}{36y^{6}}\right)=\\ \\ \\ \frac{-432x^{9}y^{6}\cdot 25x^{2}}{25y^{12}x^{4}\cdot 36y^{6}}=\frac{-10800x^{11}y^{6}}{900x^{4}y^{18}}=-12\frac{x^{7}}{y^{12}}=-12x^{7}y^{-12}\\](http://s0.wp.com/latex.php?latex=+++%5Cleft+%5B+%5Cfrac%7B-432x%5E%7B9%7Dy%5E%7B6%7D%7D%7B25y%5E%7B12%7Dx%5E%7B4%7D%7D%5Cright+%5D%3A%5Cleft%28+%5Cfrac%7B36y%5E%7B6%7D%7D%7B25x%5E%7B2%7D%7D%5Cright%29%3D%5Cleft+%5B+%5Cfrac%7B-432x%5E%7B9%7Dy%5E%7B6%7D%7D%7B25y%5E%7B12%7Dx%5E%7B4%7D%7D%5Cright+%5D+%5Ccdot+%5Cleft%28+%5Cfrac%7B25x%5E%7B2%7D%7D%7B36y%5E%7B6%7D%7D%5Cright%29%3D%5C%5C+++%5C%5C+%5C%5C+++%5Cfrac%7B-432x%5E%7B9%7Dy%5E%7B6%7D%5Ccdot+25x%5E%7B2%7D%7D%7B25y%5E%7B12%7Dx%5E%7B4%7D%5Ccdot+36y%5E%7B6%7D%7D%3D%5Cfrac%7B-10800x%5E%7B11%7Dy%5E%7B6%7D%7D%7B900x%5E%7B4%7Dy%5E%7B18%7D%7D%3D-12%5Cfrac%7Bx%5E%7B7%7D%7D%7By%5E%7B12%7D%7D%3D-12x%5E%7B7%7Dy%5E%7B-12%7D%5C%5C+++&bg=ffffff&fg=000000&s=2 "\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ]:\left( \frac{36y^{6}}{25x^{2}}\right)=\left [ \frac{-432x^{9}y^{6}}{25y^{12}x^{4}}\right ] \cdot \left( \frac{25x^{2}}{36y^{6}}\right)=\\ \\ \\ \frac{-432x^{9}y^{6}\cdot 25x^{2}}{25y^{12}x^{4}\cdot 36y^{6}}=\frac{-10800x^{11}y^{6}}{900x^{4}y^{18}}=-12\frac{x^{7}}{y^{12}}=-12x^{7}y^{-12}\\")
Im letzten Schritt wurde entsprechend gekürzt.
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