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 \(\blue{a}^\orange{0}=1\).
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Examples
\[\begin{array}{rcl} \blue x^\orange0 &=& 1 \\ \blue x^\orange1 &=& \blue x \\ \blue{x}^\orange{2} & = &\blue{x} \cdot \blue{x} \\ \blue{x}^{\orange{3}} &=& \blue{x} \cdot \blue{x} \cdot \blue{x} \\ \blue{x}^\orange{4} &=& \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \end{array} \]
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We remember this notation from numbers. There we have:
\[\blue{3} \cdot \blue{3} \cdot \blue{3} \cdot \blue{3} = \blue{3}^\orange 4\]
Hence, with variables, we have:
\[\blue{x}\cdot \blue{x}\cdot \blue{x}\cdot \blue{x}= \blue{x}^\orange 4\]
Above, the powers for non-negative integer exponents like #\blue x^\orange 1# and #\blue x^\orange 2# are defined. But what does it mean to have a negative exponent? For example, wat does #\blue x^{\orange{-3}}# mean?
For integer #\orange n > 0# and #\blue a \ne 0# we define:
\[\blue{a}^{-\orange{n}}=\dfrac{1}{\blue{a}^\orange{n}}\]
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Examples
\[\begin{array}{rcl}\blue{x}^{-\orange{1}}&=& \dfrac{1}{\blue{x}^\orange{1}} \\ \blue{x}^{-\orange{2}}&=& \dfrac{1}{\blue{x}^\orange{2}}\\ \blue{x}^{-\orange{3}}&=& \dfrac{1}{\blue{x}^\orange{3}} \end{array}\]
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The rule also holds for integer #\orange n \leq 0#:
\[\blue{a}^{-\orange{n}}=\dfrac{1}{\blue{a}^\orange{n}}\]
In general, the expression holds for all integer #\orange n#.
Example
\[\blue{x}^{3}=\blue{x}^{-\orange {-3}}=\dfrac{1}{\blue{x}^\orange{-3}}\]
With this, we have defined #\blue a^\orange n# for every integer #\orange n#.
Write #x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x# as a power.
#x^{8}#
Since #x# is multiplied by itself exactly #8# times, we have #x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x=# #x^{8}#.