• Potenzrechenregeln
  • Simon Brückner
  • 18.04.2020
  • Mathematik
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  • Potenzrechenregeln
    Potenz

    Für nN\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} n\in\mathbb{N} erhält man an\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^n indem man a\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a n-mal mit sich selbst multipliziert:
    an=aanmal\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^n=\underbrace{a\cdot\dots\cdot a}_{n-mal}

    Bsp.:
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}1) 23=222=8\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^3=2\cdot2\cdot2=8
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}2) 32=33\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^2=3\cdot3
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}3) (3)2=(3)(3)=9\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-3)^2=(-3)\cdot(-3)=9
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}4) 0,24=0,0016\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,2^4=0,0016
    Potenzen binden stärker als Rechen- und Vorzeichen:
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}5) 32=33=9\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} -3^2=-3\cdot 3=-9
    \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}6) 342=316=48\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3\cdot4^2=3\cdot 16=48

    Berechnen Sie im Kopf! Vereinfachen Sie so weit wie möglich!
    • 42=16\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^2=\cloze{16}
    • 82=64\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 8^2=\cloze{64}
    • 44=64\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^4=\cloze{64}
    • (5)2=25\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-5)^2=\cloze{25}
    • 52=25\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} -5^2=\cloze{-25}
    • (5)3=125\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-5)^3=\cloze{\small-125}
    • 0,53=0,125\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,5^3=\cloze{\small0,125}
    • 503=125000\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 50^3=\cloze{\small125000}
    • (25)3=252525=8125\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{2}{5})^3=\cloze{\small\textstyle\frac{2}{5}\cdot\frac{2}{5}\cdot\frac{2}{5}=\frac{8}{125}}

    Tipp: Nutzen Sie eine Merkhilfe, wenn Sie sich nicht sicher sind:

    Für a,bR\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a, b\in\mathbb{R} und n,mN\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} n, m\in\mathbb{N} gelten bestimmte Potenzrechenregeln. Ergänzen Sie diese!
    • (A) anam=an+m\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^n\cdot a^m=\cloze{a^{n+m}}
    • (B) anbn=(ab)n\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^n\cdot b^n=\cloze{(a\cdot b)^n}
    • C) (an)m=anm\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (a^n)^m=\cloze{a^{n\cdot m}}
    • D) an=1an\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^{-n}=\cloze{\textstyle\frac{1}{a^n}}
    • E) a1n=an\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^{\frac{1}{n}}=\cloze{\sqrt[n]{a}}
    Notieren Sie jeweils auf dem Gleichheitszeichen, welche Regel Sie anwenden und ergänzen Sie die Beispiele!
    • 32=(D)19\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^{-2}\cloze{\textstyle\overset{(D)}{=}\frac{1}{9}}
    • 16x84=(H)2x2\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[4]{16x^8}\cloze{\overset{(H)}{=}2x^2}
    • (23)3=(G)827\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{2}{3})^3\cloze{\textstyle\overset{(G)}{=}\frac{8}{27}}
    • a3a6=(A)a9\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^3\cdot a^6\cloze{\overset{(A)}{=}a^9}
    • (x2)4=(C)x8\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (x^2)^4\cloze{\overset{(C)}{=}x^8}
    • 3x2x=(B)6x\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^x\cdot 2^x\cloze{\overset{(B)}{=}6^x}
    • 813=(E)83=2\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 8^\frac{1}{3}\cloze{\small\overset{(E)}{=}\sqrt[3]{8}=2}
    • 54:52=(I)52=25\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 5^4:5^2\cloze{\overset{(I)}{=}5^2=25}
    • 64x83=(H)4x83=(E)/(C)4x83\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[3]{64x^8}\cloze{\small\overset{(H)}{=}4\sqrt[3]{x^8}\overset{(E)/(C)}{=}4x^\frac{8}{3}}
    Aus diesen Regeln kann man weitere Regeln herleiten:
    • F) a0=1\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^0=\cloze{1}
      a1=a\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{8pt}a^1=\cloze{a}
    • G) (ab)n=anbn\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{a}{b})^n=\cloze{\textstyle\frac{a^n}{b^n}}
    • H) abn=anbn\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[n]{a\cdot b}=\cloze{\sqrt[n]{a}\cdot\sqrt[n]{b}}
    • I) an:am=anm\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} a^n:a^m=\cloze{a^{n-m}}
  • 1
    Berechnen Sie ohne Hilfsmittel!
    • 22\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^2
    • (3)4\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-3)^4
    • 45\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^5
    • 0,82\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,8^2
    • 23\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^3
    • 0,92\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,9^2
    • (2)5\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-2)^5
    • (2)3\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-2)^3
    • 1,32\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 1,3^2
    • (3)3\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-3)^3
    • 42\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^2
    • (2)2\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (-2)^2
    Quadratzahlen

    12=1112=121\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 1^2=1\hspace{22pt} 11^2=121
    22=4122=144\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 2^2=4\hspace{22pt} 12^2=144
    32=9132=169\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 3^2=9\hspace{22pt} 13^2=169
    42=16142=196\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 4^2=16\hspace{17pt} 14^2=196
    52=25152=225\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 5^2=25\hspace{17pt} 15^2=225
    62=36162=256\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 6^2=36\hspace{17pt} 16^2=256
    72=49172=289\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 7^2=49\hspace{17pt} 17^2=289
    82=64182=324\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 8^2=64\hspace{17pt} 18^2=324
    92=81192=361\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt} 9^2=81\hspace{17pt} 19^2=361
    102=100202=400\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 10^2=100\hspace{12pt} 20^2=400

    • 53=1125\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 5^{-3}\cloze{\textstyle=\frac{1}{125}}
    • 42=116\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^{-2}\cloze{\textstyle=\frac{1}{16}}
    • (14)2=16\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{4})^{-2}\cloze{\textstyle=16}
    • (15)2=25\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{5})^{-2}\cloze{\textstyle=25}
    • 43=164\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^{-3}\cloze{\textstyle=\frac{1}{64}}
    • (12)2=4\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{2})^{-2}\cloze{\textstyle=4}
    • (12)3=8\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{2})^{-3}\cloze{\textstyle=8}
    • (14)4=256\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{4})^{-4}\cloze{\textstyle=256}
    • 31=13\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^{-1}\cloze{\textstyle=\frac{1}{3}}
    • 41=14\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 4^{-1}\cloze{\textstyle=\frac{1}{4}}
    • (14)3=64\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{4})^{-3}\cloze{\textstyle=64}
    • (15)1=5\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{1}{5})^{-1}\cloze{\textstyle=5}
    Kubikzahlen

    13=1113=1331\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}1^3=1\hspace{22pt} 11^3=1331
    23=8123=1728\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}2^3=8\hspace{22pt} 12^3=1728
    33=27133=2197\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}3^3=27\hspace{17pt} 13^3=2197
    43=64143=2744\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}4^3=64\hspace{17pt} 14^3=2744
    53=125153=3375\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}5^3=125\hspace{12pt} 15^3=3375
    63=216163=4096\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}6^3=216\hspace{12pt} 16^3=4096
    73=343173=4913\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}7^3=343\hspace{12pt} 17^3=4913
    83=512183=5832\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}8^3=512\hspace{12pt} 18^3=5832
    93=729193=6859\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{5pt}9^3=729\hspace{12pt} 19^3=6859
    103=1000203=8000\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 10^3=1000\hspace{7pt} 20^3=8000

    2
    Erklären Sie, welche häufigen Fehler hier jeweils gemacht wurden und korrigieren Sie.
    • 23=8\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^{-3}=-8
      \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \textrm{}
    • 4x6=2x6\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt{4x^6}=2x^6
    • 2324=47\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^3\cdot 2^4=4^7
      \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \textrm{}
    • (43)2=49\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (4^3)^2=4^9
    • 23=6\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^3=6
      \gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \textrm{}
    • 2324=212\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 2^3\cdot 2^4=2^{12}
    3
    Berechnen Sie im Kopf!
    • 3262=182=324\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^2\cdot 6^2\cloze{=18^2=324}
    • 35:33=353=32=9\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 3^5:3^3\cloze{=3^{5-3}=3^2=9}
    • 122=1122=1144\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 12^{-2}\cloze{\textstyle=\frac{1}{12^2}=\frac{1}{144}}
    • 0,043=(4100)3\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,04^3\cloze{\textstyle=\left(\frac{4}{100}\right)^3}
      =641000000=0,000064\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \cloze{\textstyle=\frac{64}{1000000}=0,000064}
    • 0,250,5=141=(12)1=2\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} 0,25^{-0,5}\cloze{\small\textstyle=\sqrt{\frac{1}{4}}^{-1}=\left(\frac{1}{2}\right)^{-1}=2}
    • 1363=1363=132=169\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[3]{13^6}\cloze{\textstyle=13^{\frac{6}{3}}=13^2=169}
    • (34)2=3242=916\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} (\frac{3}{4})^2\cloze{\textstyle=\frac{3^2}{4^2}= \frac{9}{16}}
    4
    Auch wenn man einen Ausdruck nicht komplett ohne Wurzel schreiben kann, kann es helfen, die Wurzel teilweise zu ziehen.
    Bsp: 722=9422=3222=622=52\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt{72}-\sqrt{2}=\sqrt{9\cdot4\cdot2}-\sqrt{2}=3\cdot2\cdot\sqrt{2}-\sqrt{2}=6\sqrt{2}-\sqrt{2}=5\sqrt{2}

    Vereinfachen Sie entsprechend!
    • 543=2273=323\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[3]{54}\cloze{\textstyle=\sqrt[3]{2\cdot27}=3\sqrt[3]{2}}
    • 12+2=24+2\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt{\frac{1}{2}}+\sqrt{2}\cloze{\small\textstyle=\sqrt{\frac{2}{4}}+\sqrt{2}} =22+2=322\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \cloze{\textstyle=\frac{\sqrt{2}}{2}+\sqrt{2}=\frac{3}{2}\sqrt{2}}
    • 432=223212=256\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \sqrt[3]{4}\cdot\sqrt{2}\cloze{\textstyle=2^\frac{2}{3}\cdot2^\frac{1}{2}=2^\frac{5}{6}}

    Zum Weiterlesen:\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{3cm}
    Primfaktorzerlegung\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{2.6cm}
    https://youtu.be/6gXQUCzv9hM\gdef\cloze#1{\colorbox{none}{\color{transparent}{\large{$\displaystyle #1$}}}} \hspace{0.95cm}