# Exponents

To find rules for working with exponents, see: Rules of Exponents

When dealing with positive integers, **exponents** can be thought of as a shorthand notation for "repeated multiplication" of a number by itself several times. For example, the shorthand for multiplying 5 copies of the number 2 is

\[ 2 \times 2 \times 2 \times 2 \times 2 = 2 ^ 5. \]

In words, we say that "\( 2^ 5 \) is \(2\) to the fifth power." In this example, 2 is the *base* and 5 is the *exponent*.

Note that neither the base nor the exponent need to be positive integers, meaning the "repeated multiplication" idea breaks down; \( 2^{-5} ,\) \( \frac{2}{3}^\frac{1}{3} ,\) and \( 0.99^{0.99} \) are all valid exponent expressions. Their nature can be worked out via the Rules of Exponents.

## What is \( 3^4\)?

We have

\[ 3 ^ 4 = 3 \times 3 \times 3 \times 3 = 9 \times 3 \times 3 = 27 \times 3 = 81 . \]

Thus \( 3^4 = 81 \). \( _\square \)

## What is \( 2^5\)?

\( 1 \times 2 = 2 \)

\( 2 \times 2 = 4 \)

\( 4 \times 2 = 8 \)

\( 8 \times 2 = 16 \)

\( 16 \times 2 = 32 \)Therefore, \( 2^ { 5 } = 32 \). \( _ \square \)

## What is \( 2^ {10} \)?

For larger powers, we have to be careful as we multiply out these terms. If you are uncertain, it is best to list out the powers in order. For example, we have

\[ \begin{array} { l | l } n & 2^n \\ \hline 1 & 2^1 = 2 \\ 2 & 2 \times 2 = 4 \\ 3 &2 \times 4 = 8 \\ 4 &2 \times 8 = 16 \\ 5 &2 \times 16 = 32 \\ 6 &2 \times 32 = 64 \\ 7 &2 \times 64 = 128 \\ 8 &2 \times 128 = 256 \\ 9 &2 \times 256 = 512 \\ 10 &2 \times 512 = 1024 \\ \end{array} \]

Hence, \( 2^ { 10 } = 1024 \). \( _ \square \)

Just like the multiplication tables, after a while you will start to be familiar with some of these numbers, and can remember what they are without having to work them out every single time. It just takes some practice.

Click here to learn more about the rules of exponents

The inverse operation of an exponential function is a logarithm. Learn more about them here.