Repeated multiplication of a variable with itself can also be written as a power:

\[\begin{array}{rcl}
\blue{a}^\orange{n} & =& \underbrace{\blue{a} \cdot \blue{a} \cdot \ldots \cdot \blue{a}}_{\orange{n}\textrm{ times}} \end{array}\]

We call #\blue{a}^\orange{n}# the #\orange{n}#-th power of #\blue{a}#.
The number #\blue{a}# is called the base, and #\orange{n}# is the exponent.

Next to that, we have \(\blue{a}^\orange{0}=1\).

Examples

\[\begin{array}{rcl}
\blue x^\orange0 &=& 1 \\
\blue x^\orange1 &=& \blue x \\
\blue{x}^\orange{2} & = &\blue{x} \cdot \blue{x} \\ \blue{x}^{\orange{3}} &=& \blue{x} \cdot \blue{x} \cdot \blue{x} \\ \blue{x}^\orange{4} &=& \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \end{array} \]

For integer #\orange n > 0# and #\blue a \ne 0# we define:

\[\blue{a}^{-\orange{n}}=\dfrac{1}{\blue{a}^\orange{n}}\]

Examples

\[\begin{array}{rcl}\blue{x}^{-\orange{1}}&=& \dfrac{1}{\blue{x}^\orange{1}} \\
\blue{x}^{-\orange{2}}&=& \dfrac{1}{\blue{x}^\orange{2}}\\
\blue{x}^{-\orange{3}}&=& \dfrac{1}{\blue{x}^\orange{3}} \end{array}\]