# Exponents

The laws of exponents, unlike laws of motion, are in no particular order

$1.\;a^m \times a^n = a^{m+n}$
$2. \; \frac{a^m}{a^n} = a^{m-n}$
$3. \; a^m \times b^m = (a\times b)^m$
$4.\; a^{\frac{n}{m}} = \sqrt[m]{a^n}$
$5. \; a^{-m} = \frac{1}{a^m}$
$6.\; a^0 = 1$
$7. \; 0^m = 0$

Law 6 and Law 7 create a conflict when $0^0$, as both give different results and both are true, so $\therefore 0^0 =$ undefined

$8. \sqrt[2m]{a} \neq undefined \to a \geq 0$

Law 8 means that negative numbers cannot have even roots

$9. \; {(a^m)}^n = a^{mn}$
$10. \; {\frac{a^m}{b^m}} = (\frac{a}{b})^m$