### Algebra: Calculating with exponents and roots

### Integer exponents

Repeated multiplication of a variable with itself can also be written as a power: \[\begin{array}{rcl} We call #\blue{a}^\orange{n}# the #\orange{n}#-th power of #\blue{a}#. Next to that, we have \(\blue{a}^\orange{0}=1\). |
\[\begin{array}{rcl} |

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: \[\blue{a}^{-\orange{n}}=\dfrac{1}{\blue{a}^\orange{n}}\] |
\[\begin{array}{rcl}\blue{x}^{-\orange{1}}&=& \dfrac{1}{\blue{x}^\orange{1}} \\ |

With this, we have defined #\blue a^\orange n# for every integer #\orange n#.

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}#.

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