Exponentiation is a mathematical operation that raises a base number to a given power, or exponent. It is a shorthand for repeated multiplication and is a key concept in algebra, calculus, and many applied sciences. The exponent tells how many times to multiply the base by itself.
\[ a^m = \underbrace{a \cdot a \cdot \dots \cdot a}_{m \text{ times}} \]
\[ \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0) \]
\[ a^m \cdot a^n = a^{m+n} \]
\[ (a \cdot b)^m = a^m \cdot b^m \]
\[ (a^m)^n = a^{mn} \]
\[ a^0 = 1 \quad (a \neq 0) \]
\[ a^{-m} = \frac{1}{a^m} \quad (a \neq 0) \]
\[ a^{\frac{1}{n}} = \sqrt[n]{a} \quad (a \geq 0 \text{ for real roots}) \]
\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad (a \geq 0 \text{ for real roots}) \]