Fórmulas Potenciação

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Potenciação

1. Nomenclatura

$a^{x}=b$

2. Definições

$a^{n}=\underbrace{a.a.a. \hspace{0.4em}... \hspace{0.4 em} . a}_{n \ termos}$

$a^{0} = 1 \quad$ $(\forall \hspace{0.3em} a \in \mathbb{R})$

$a^{1}=a$

$a^{n} = a^{n-1}.a \quad$ $(\forall \ a \in \mathbb{R} \hspace{0.5em} \wedge \hspace{0.5em} \forall \ n \in \mathbb{N})$

3. Propriedades

$a^{m}.a^{n} = a^{m+n}$

$\dfrac{a^{m}}{a^{n}} = a^{m-n} \quad$ $(a \neq 0)$

$(a^{m})^{n} = a^{m.n}$

$\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^n} \quad$ $(b \neq 0)$

$(a)^{-n} = \dfrac{1}{a^n} \quad$ $(a \neq 0)$

$(a.b)^{n} = a^{n}.b^{n}$

$a^{m}.b^{m} = (a.b)^{m}$

$(a)^{\dfrac{m}{n}} = {\sqrt[\style{font-size:140%;}{n}]{a^m}}$

$\left(\dfrac{a}{b} \right)^{n}= \left( \dfrac{b}{a} \right) ^{-n}$

4. Observações

$(a+b)^{n} \hspace{0.5em} \style{color:Crimson; font-weight:bold;}{\neq} \hspace{0.5em} a^{n} . b^{n}$

$(a-b)^{n} \hspace{0.5em} \style{color:Crimson; font-weight:bold;}{\neq} \hspace{0.5em} a^{n}-b^{n}$