Puteri cu exponent rațional

$$\begin{gathered} Fie a\ge 0, \frac{m}{n}\in \mathrm{\mathbb{Q}}, n\ge 2. \\ \boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}} \\ \bullet Dacă m=1 \Rightarrow \boxed{ a^{\frac{1}{n}}=\sqrt[n]{a}} \end{gathered}$$

 

Proprietățile puterilor cu exponent rațional (a, b >0):

 

$$\begin{gathered} 1. a^{\frac{m}{n}}·a^{\frac{p}{q}}=a^{\frac{m}{n}+\frac{p}{q}} \\ 2. \frac{a^{{\displaystyle \frac{m}{n}}}}{a^{{\displaystyle \frac{p}{q}}}}=a^{\frac{m}{n}-\frac{p}{q}} \\ 3. {\left(a·b\right)}^{\frac{m}{n}}=a^{\frac{m}{n}}·b^{\frac{m}{n}} \\ 4. {\left(\frac{a}{b}\right)}^{\frac{m}{n}}=\frac{a^{{\displaystyle \frac{m}{n}}}}{b^{{\displaystyle \frac{m}{n}}}} \\ 5. {\left(a^{\frac{m}{n}}\right)}^{\frac{p}{q}}=a^{\frac{m}{n}·\frac{p}{q}}. \end{gathered}$$