Potenzgesetze

$2^3\cdot 2^2=2^{3+2}\backslash ;\backslash ;2^3\backslash cdot\; 2^2=2^\{3+2\}$ $\backslash \backslash$ $=(2\cdot 2\cdot 2)\cdot (2\cdot 2)=2\cdot 2\cdot 2\cdot 2\cdot 2=2^5=(2\backslash cdot\; 2\backslash cdot\; 2)\backslash cdot\; (2\backslash cdot\; 2)\backslash \backslash \; =\; 2\backslash cdot\; 2\backslash cdot\; 2\backslash cdot\; 2\backslash cdot\; 2\; =\; 2^\{5\}$

$a^x\cdot a^y=a^{x+y}\backslash ;\backslash ;a^x\backslash cdot\; a^y=a^\{x+y\}$

Multiplikation bei $\backslash \backslash \backslash \backslash$gleicher Basis $aa$

$\frac{2^3}{2^2}=2^{3-2}\backslash ,\backslash ,\backslash frac\{2^3\}\{2^2\}=2^\{3-2\}$ $\backslash \backslash$ $=\frac{2\cdot 2\cdot 2}{2\cdot 2}=2^1=2^{3-2}=\backslash frac\{2\backslash cdot2\backslash cdot2\backslash ;\}\{2\backslash cdot2\}\backslash \backslash \; =\; 2^1\; =\; 2^\{3-2\}$

$\frac{a^x}{a^y}=a^{x-y}\backslash ,\backslash ,\backslash frac\{a^x\}\{a^y\}=a^\{x-y\}$

Division bei $\backslash \backslash \backslash \backslash$gleicher Basis $aa$

$2^3\cdot 3^3={(2\cdot 3)}^3\backslash ;\backslash ;2^3\backslash cdot\; 3^3=\backslash left(2\backslash cdot\; 3\backslash right)^3$ $\backslash \backslash$ $=(2\cdot 2\cdot 2)\cdot (3\cdot 3\cdot 3)=(2\cdot 3)\cdot (2\cdot 3)\cdot (2\cdot 3)=(2\cdot 3{)}^3=\; (\; 2\backslash cdot\; 2\backslash cdot\; 2\; )\; \backslash cdot\; (\; 3\backslash cdot\; 3\backslash cdot\; 3\; )\backslash \backslash \; =\; (2\; \backslash cdot\; 3)\; \backslash cdot\; (2\; \backslash cdot\; 3)\; \backslash cdot\; (2\; \backslash cdot\; 3)\backslash \backslash \; =\; (2\; \backslash cdot\; 3)^3$

$a^x\cdot b^x={(a\cdot b)}^x\backslash ;\backslash ;a^x\backslash cdot\; b^x=\backslash left(a\backslash cdot\; b\backslash right)^x$

Multiplikation bei $\backslash \backslash \backslash \backslash$gleichem Exponenten $xx$

$\frac{2^3}{3^3}={\left(\frac{2}{3}\right)}^3\backslash frac\{2^3\}\{3^3\}=\backslash left(\backslash frac\; 2\; 3\backslash right)^3$ $\backslash \backslash$ $=\frac{2\cdot 2\cdot 2}{3\cdot 3\cdot 3}=\frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3}={\left(\frac{2}{3}\right)}^3=\; \backslash frac\{2\backslash cdot\; 2\backslash cdot\; 2\}\{3\backslash cdot\; 3\backslash cdot\; 3\}\backslash \backslash =\; \backslash frac\; 2\; 3\; \backslash cdot\; \backslash frac\; 2\; 3\; \backslash cdot\; \backslash frac\; 2\; 3\backslash \backslash =\; \backslash left(\backslash frac\; 2\; 3\backslash right)^3$

$\frac{a^x}{b^x}={\left(\frac{a}{b}\right)}^x\backslash frac\{a^x\}\{b^x\}=\backslash left(\backslash frac\{a\}\{b\}\backslash right)^x$

Division bei $\backslash \backslash \backslash \backslash$gleichem Exponenten $xx$

${(-2)}^6=2^6\backslash left(-2\backslash right)^6=2^6$ $\backslash \backslash$ $=(-2{)}^2\cdot (-2{)}^2\cdot (-2{)}^2=2^2\cdot 2^2\cdot 2^2=2^6=(-2)^2\backslash cdot(-2)^2\backslash cdot(-2)^2\backslash \backslash =2^2\backslash cdot2^2\backslash cdot2^2\backslash \backslash =\; 2^6$

${(-a)}^x=a^x\backslash left(-a\backslash right)^x=a^x$

Negative Basis und $\backslash \backslash \backslash \backslash$gerader Exponent.

${(-2)}^5=-(2^5)\backslash left(-2\backslash right)^5=-(2^5)$ $\backslash \backslash$ $=(-2{)}^2\cdot (-2{)}^2\cdot (-2)=2^2\cdot 2^2\cdot (-2)=-(2^5)=(-2)^2\backslash cdot(-2)^2\backslash cdot(-2)\backslash \backslash =2^2\backslash cdot2^2\backslash cdot(-2)\backslash \backslash =\; -(2^5)$

$\text{}{(-a)}^x=-(a^x)\backslash left(-a\backslash right)^x=-(a^x)$

Negative Basis und $\backslash \backslash \backslash \backslash$ungerader Exponent.

$2^0=12^0=1$ $\backslash \backslash$ $2^2=2\cdot 2=2\cdot 2\cdot 12^1=2=2\cdot 12^0=12^2\; =\; 2\; \backslash cdot\; 2\; =\; 2\; \backslash cdot\; 2\; \backslash cdot\; 1\; \backslash \backslash \; 2^1=\; 2\; =\; 2\; \backslash cdot\; 1\; \backslash \backslash \; 2^0=\; 1$

$a^0=1a^0=1$

Null im Exponenten $\backslash \backslash \backslash \backslash$bei beliebiger Basis $a\mathrm{\ne }0a\backslash neq\; 0$

$2^{-3}=\frac{1}{2^3}2^\{-3\}=\backslash frac\{1\}\{2^3\}$ $\backslash \backslash$ $1=2^0=2^{3+(-3)}=2^3\cdot 2^{-3}1=2^0=2^\{3+(-3)\}\backslash \backslash =2^3\backslash cdot\; 2^\{-3\}\backslash \backslash$ Damit dies $11$ ist, muss $\backslash \backslash$ $2^{-3}=\frac{1}{2^3}2^\{-3\}=\backslash frac1\{2^3\}\backslash \backslash$ gelten, denn $\backslash \backslash$ $2^3\cdot \frac{1}{2^3}=\frac{2^3}{2^3}=12^3\backslash cdot\backslash frac1\{2^3\}=\backslash frac\{2^3\}\{2^3\}=1$

$a^{-x}=\frac{1}{a^x}a^\{-x\}=\backslash frac1\{a^x\}$

Negative Exponenten

$2^{\frac{1}{3}}=\sqrt[3]{2}2^\backslash frac\; 1\; 3\; =\backslash sqrt[3]2$ $\backslash \backslash \backslash \backslash$ $2=2^1=2^{\frac{1}{3}\cdot 3}={\left(2^{\frac{1}{3}}\right)}^{3}2=2^1=2^\{\backslash frac13\; \backslash cdot\; 3\}\backslash \backslash =\backslash left(2^\{\backslash frac13\}\backslash right)^3\backslash \backslash$ Damit dies $22$ ist, muss $\backslash \backslash \backslash \backslash$ $2^{\frac{1}{3}}=\sqrt[3]{2}2^\{\backslash frac13\}=\backslash sqrt[3]2$ gelten, $\backslash \backslash \backslash \backslash$ denn ${\left(\sqrt[3]{2}\right)}^3=2\backslash left(\backslash sqrt[3]\{2\}\backslash right)^3=2$

$a^{\frac{1}{n}}=\sqrt[n]{a}a^\backslash frac1n=\backslash sqrt[n]a$

Einheitsbrüche $\backslash \backslash \backslash \backslash$im Exponenten