# Index Notation

A convenient way to write a product of $\textit{identical factors}$ is to use $\textbf{exponential}$ or $\textbf{index notation}$.
Rather than writing $5 \times 5 \times 5 \times 5$, we can write this product as $5^4$.

The small $4$ is called the $\textbf{exponent}$ or $\textbf{index}$, and the $5$ is called the $\textbf{base}$.
If $n$ is a positive integer, then $a^n$ is the product of $n$ factors of $a$.
$$a^n=\overbrace {a \times a \times a \times \cdots \times a}^{n}$$
We say that $a$ is the $\textbf{base}$, and $n$ is the $\textbf{exponent}$ or $\textbf{index}$.

The following table shows the first four powers of $2$.

$$\begin{array}{|c|c|c|c|} \hline \textit{Natural number} & \textit{Factorised form} & \textit{Exponent form} & \textit{Spoken form} \\ \hline 2 & 2 & 2^1 & \text{two} \\ \hline 4 & 2 \times 2 & 2^2 & \text{two squared} \\ \hline 8 & 2 \times 2 \times 2 & 2^3 & \text{two cubed} \\ \hline 16 & 2 \times 2 \times 2 \times 2 & 2^4 & \text{two to the fourth} \\ \hline \end{array}$$

Any non-zero number raised to the power zero is equal to $1$.
$$a^0 = 1, a \ne 0$$
$0^0$ is undefined.

$\textbf{Negative Bases}$

$\begin{eqnarray} (-1)^1 &=& -1 \\ (-1)^2 &=& -1 \times -1 = 1 \\ (-1)^3 &=& -1 \times -1 \times -1 = -1 \\ (-1)^4 &=& -1 \times -1 \times -1 \times -1 = 1 \\ (-2)^1 &=& -2 \\ (-2)^2 &=& -2 \times -2 = 4 \\ (-2)^3 &=& -2 \times -2 \times -2 = -8 \\ (-2)^4 &=& -2 \times -2 \times -2 \times -2 = 16 \\ \end{eqnarray}$

From the patterns above we can see that:

$\textbf{negative}$ base raised to an $\textbf{even}$ exponent is $\textbf{positive}$.

$\textbf{negative}$ base raised to an $\textbf{odd}$ exponent is $\textbf{negative}$.

## Example 1

List the first four powers of 3.

$3^1 = 3$
$3^2 = 3 \times 3 = 9$
$3^3 = 3 \times 3 \times 3 = 27$
$3^4 = 3 \times 3 \times 2 \times 3 = 81$

## Example 2

Simplify $(-1)^{17}$.

$(-1)^{17} = -1$

## Example 3

Simplify $(-1)^{18}$.

$(-1)^{18} = 1$

## Example 4

Simplify $5^0$.

$5^0 = 1$ 