Puterea cu exponent întreg a unui numar real

Fie a un număr real, iar n un număr natural nenul.

$$\begin{gathered} a^n=\underbrace{a·a·a·...·a} \\ n ori \end{gathered}$$

a- se numește bază

n- se numește exponent

Exemplu:

${\left(\sqrt{2}\right)}^3=\sqrt{2}·\sqrt{2}·\sqrt{2}=2\sqrt{2}$

$$\begin{gathered} \boxed{\overline{)\overline{) {\left(\sqrt{a}\right)}^n=\sqrt{a^n} \\ {\left(a\sqrt{b}\right)}^n=a^n\sqrt{b^n} }}} (a, b\in {\mathrm{ℝ}}^{*}, b>0, n\in {\mathrm{\mathbb{N}}}^{*}) \end{gathered}$$

 

Reguli de calcul cu puteri

$a^m·a^n=a^{m+n} (a\in {\mathrm{ℝ}}^{*}, m, n\in \mathrm{\mathbb{N}})$

$a^m:a^n=a^{m-n} (a\in {\mathrm{ℝ}}^{*}, m, n\in \mathrm{\mathbb{N}}, m\ge n)$

${\left(a^m\right)}^n=a^{m·n} (a\in {\mathrm{ℝ}}^{*}, m, n\in \mathrm{\mathbb{N}})$

$a^m·b^m=(a·b{)}^m (a,b\in {\mathrm{ℝ}}^{*}, m\in \mathrm{\mathbb{N}})$

$a^m:b^m=(a:b{)}^m (a,b\in {\mathrm{ℝ}}^{*}, m\in \mathrm{\mathbb{N}})$

$(-a{)}^n=\left\{\begin{array}{l} a^n, n-par\\ -a^n, n-impar\end{array}\right.a\in {\mathrm{ℝ}}^{*}, n\in \mathrm{\mathbb{N}}$

$a^0=1 (a\in {\mathrm{ℝ}}^{*})$

$a^1=a (a\in {\mathrm{ℝ}}^{*})$

Dacă exponentul este negativ, atunci vom aplica următoarea formulă:

$\boxed{a^{-n}=\frac{1}{a^n}} (a\in {\mathrm{ℝ}}^{*}, n\in \mathrm{\mathbb{N}})$

În particular, dacă n = 1, avem:

$\boxed{a^{-1}=\frac{1}{a}} (a\in {\mathrm{ℝ}}^{*}, n\in \mathrm{\mathbb{N}}).$

Exemple:

$$\begin{gathered} {\left(\sqrt{3}\right)}^{-4}=\frac{1}{{\left(\sqrt{3}\right)}^4}=\frac{1}{9} \\ {\left(2\sqrt{5}\right)}^{-1}=\frac{1}{2\sqrt{5}}. \end{gathered}$$