Exponentiation

Integer Exponents

• $a^n=\underset{n\text{ times}}{\underbrace{a\times a\times \cdots \times a\times a}}$

• Exponents Rules

  • $a^{n+m}=a^n\cdot a^m$
  • $(a^n{)}^m=a^{nm}$
  • ${(ab)}^n=a^n{b}^n$
  • $1^n=1$
  • $a^1=a$
  • $a^0=1$
  • $a>1\wedge n>m⟹a^n>a^m$
  • $a^{n-m}=\frac{a^n}{a^m}$
  • ${(\frac{a}{b})}^n=\frac{a^n}{b^n}$
  • ${(\frac{a}{b})}^{-n}={(\frac{b}{a})}^n$
  • ${\log }_b{b}^x=x$ (where $0<b\ne 1$)

Order of operations $a^{n^m}=a^{(n^m)}$

Rational Exponents

• $a^{m/n}:=\sqrt[n]{a^m}$

• The exponents rules also hold for rational exponents

Real Exponents

Assumption $r\in \mathbb{R},a>0,\underset{n\to \infty }{\lim }{q}_n=r$

• Definition - $a^r:=\underset{n\to \infty }{\lim }{a}^{q_n}$

• The exponents rules also hold for real exponents

Domain of $f(x)=x^r$

todo

	$r<0$	$r>0$
$r\in \mathbb{Z}$ or $r=\frac{p}{q}\in \mathbb{Q}$ and $q$ is odd, (OR, for complex solution)	$\mathbb{R}\setminus \{0\}$	$\mathbb{R}$
$r=\frac{p}{q}\in \mathbb{Q}$ and $q$ is even, OR $r\in \mathbb{R}\setminus \mathbb{Q}$	$(0,\infty )$	$[0,\infty )$