Power Laws

$\begin{array}{l}{2}^3\cdot 2^2=2^{3+2}\\ \begin{array}{rl}{2}^3\cdot 2^2& =(2\cdot 2\cdot 2)\cdot (2\cdot 2)\\ & =2\cdot 2\cdot 2\cdot 2\cdot 2\\ & =2^5\\ & =2^{3+2}\end{array}\end{array}$

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

Multiplication with the same base $a$

$\begin{array}{l}\frac{2^3}{2^2}=2^{3-2}\\ \frac{2^3}{2^2}=\frac{2\cdot 2\cdot 2}{2\cdot 2}=2^1=2^{3-2}\end{array}$

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

Division with the same base $a$

$\begin{array}{l}{2}^3\cdot 3^3={(2\cdot 3)}^3\\ \begin{array}{rl}{2}^3\cdot 3^3& =(2\cdot 2\cdot 2)\cdot (3\cdot 3\cdot 3)\\ & =(2\cdot 3)\cdot (2\cdot 3)\cdot (2\cdot 3)\\ & =(2\cdot 3{)}^3\end{array}\end{array}$

$a^x\cdot b^x={(a\cdot b)}^x$

Multiplication with the same exponent $x$

$\begin{array}{l}\frac{2^3}{3^3}={\left(\frac{2}{3}\right)}^3\\ \begin{array}{rl}\frac{2^3}{3^3}& =\frac{2\cdot 2\cdot 2}{3\cdot 3\cdot 3}\\ & =\frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3}\\ & ={\left(\frac{2}{3}\right)}^3\end{array}\end{array}$

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

Division with the same exponent $x$

$\begin{array}{l}{(-2)}^6=2^6\\ =(-2{)}^2\times (-2{)}^2\times (-2{)}^2\\ =2^2\times 2^2\times 2^2\\ =2^6\end{array}$

${(-a)}^x=a^x$

Negative base and even exponent

$\begin{array}{l}{(-2)}^5=-(2^5)\\ =(-2{)}^2\times (-2{)}^2\times (-2)\\ =2^2\times 2^2\times (-2)\\ =-(2^5)\end{array}$

${(-a)}^x=-(a^x)$

Negative base and odd exponent

$\begin{array}{l}{2}^0=1\\ 2^2=2\cdot 2=2\cdot 2\cdot 1\\ 2^1=2=2\cdot 1\\ 2^0=1\end{array}$

$a^0=1$

Zero in the exponent with base $a\ne 0$

$\begin{array}{l}{2}^{-3}=\frac{1}{2^3}\\ 1=2^0=2^{3+(-3)}\\ =2^3\cdot 2^{-3}\\ \text{For this to be 1, we need}\\ 2^{-3}=\frac{1}{2^3}\\ \text{since}\\ 2^3\cdot \frac{1}{2^3}=\frac{2^3}{2^3}=1\end{array}$

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

Negative exponent

$\begin{array}{l}{2}^{\frac{1}{3}}=\sqrt[3]{2}\\ 2=2^1=2^{\frac{1}{3}\cdot 3}\\ ={\left(2^{\frac{1}{3}}\right)}^3\\ \text{For this to be 2, we need}\\ 2^{\frac{1}{3}}=\sqrt[3]{2}\\ \text{since}\\ {\left(\sqrt[3]{2}\right)}^3=2\end{array}$

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

Unit fractions in the exponent