### Numbers: Powers and roots

### Powers

We can repeatedly multiply a number by itself. We write this as follows:

\[\begin{array}{rclrc}\blue2^\orange0&&&=&1\\\blue2^\orange1&=&\blue2&=&2\\\blue2^\orange2&=&\blue2 \times \blue2&=&4\\\blue2^\orange3&=&\blue2 \times \blue2 \times \blue2&=&8 \\ \blue2^\orange4&=&\blue2 \times \blue2 \times \blue2 \times \blue2& =&16\end{array}\]

We call #\blue2^\orange3# a **power.**

Here, we call #\blue2# the #\blue{\text{base}}# of the power. This number is multiplied by itself repeatedly.

We call #\orange3# the #\orange{\text{exponent}}#. This number indicates how many times the #\blue{\text{base}}# is multiplied by itself.

**Examples**

\[\begin{array}{rcl}\blue5^\orange4&=&\blue5 \times \blue5 \times \blue5 \times \blue5 \\ &=& 625 \\ \\ \blue3^\orange2&=&\blue3 \times \blue3 \\ &=& 9\\ \\ \blue{10}^\orange3&=&\blue{10} \times \blue{10} \times \blue{10} \\ &=&1000 \\ \\ (\blue{-4})^\orange4&=&\blue{-4} \times \blue{-4} \times \blue{-4} \times \blue{-4} \\ &=&256\end{array}\]

#\begin{array}{rcl}

(-6)^3 &=&-6 \cdot -6 \cdot -6\\ & &\phantom{xxx}\blue{\text{exponentiation is repeated multiplication}}\\

&=& -216\\

&&\phantom{xxx}\blue{\text{multiplied}}

\end{array}#

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