Exponent

Exponents are a short way to write repeated multiplication. They help us work with large numbers more easily.

Definitions

Imagine you have a chessboard. You place two grains of wheat on the first square, four grains on the second square, eight grains on the third square, and so on, doubling the number of grains on each next square.

How many grains of wheat are on the last square of a chessboard with 64 squares?

Answer

Square number	Number of grains
$1$	$2$
$2$	$2 \times 2$
$3$	$2 \times 2 \times 2$
$\vdots$	$\vdots$
$64$	$\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}$

Rather than writing $\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}$, we can write this product as $2^{64}$.
This means there are $2^{64}$ grains on the last square. Using a calculator:$$2^{64}=18\,446\,744\,073\,709\,551\,616.$$This is an enormous number!

Definition Exponentiation

Exponentiation is repeated multiplication of a number by itself.
For a number $a$ and a positive whole number $n$,$$a^n = \overbrace{a \times a \times \dots \times a}^{n\ \text{factors}}.$$

Example

Write using exponent notation: $5 \times 5 \times 5$.

Answer

$5 \times 5 \times 5 = 5^3$

Definition Vocabulary

$$\begin{array}{|c|c|c|c|}\hline\text{Value} & \text{Expanded form} & \text{Exponent notation} & \text{Spoken form} \\ \hline2 & 2 & 2^1 & 2\ \text{or}\ 2\ \text{to the power of}\ 1 \\ 4 & 2 \times 2 & 2^2 & 2\ \text{squared or}\ 2\ \text{to the power of}\ 2 \\ 8 & 2 \times 2 \times 2 & 2^3 & 2\ \text{cubed or}\ 2\ \text{to the power of}\ 3 \\ 16 & 2 \times 2 \times 2 \times 2 & 2^4 & 2\ \text{to the power of}\ 4 \\ 32 & 2 \times 2 \times 2 \times 2 \times 2 & 2^5 & 2\ \text{to the power of}\ 5 \\ \hline\end{array}$$

Example

Find the value of $2^3$.

Answer

$$\begin{aligned}[t]2^3 &= 2 \times 2 \times 2 \\ &= 8\end{aligned}$$

Exponent Law

Discover

Let's look at an example:$$\begin{aligned}\textcolor{colordef}{7}^{\textcolor{colorprop}{3}} \times \textcolor{colordef}{7}^{\textcolor{olive}{2}}&= \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{3}\,\text{factors}} \times \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{olive}{2}\,\text{factors}} \\ &= \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{3}+\textcolor{olive}{2}\,\text{factors}} \\ &= \textcolor{colordef}{7}^{\textcolor{colorprop}{3}+\textcolor{olive}{2}}.\end{aligned}$$In this example we are multiplying two powers with the same base (7).
We can see that we keep the base and add the exponents: $3 + 2 = 5$.
In general, when a number $\textcolor{colordef}{a}$ is raised to the power $\textcolor{colorprop}{m}$ and multiplied by the same number raised to the power $\textcolor{olive}{n}$, that is$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}},$$the result is equal to $\textcolor{colordef}{a}$ raised to the sum of the exponents:$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}.$$

Proposition Exponent Law 1

When we multiply two powers with the same base, we keep the base and add the exponents:$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}.$$

Proof

$$\begin{aligned}\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}&= \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{colorprop}{m}\ \text{factors}} \times \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{olive}{n}\ \text{factors}} \\ &= \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}\ \text{factors}} \\ &= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}\end{aligned}$$

Example

Simplify $5^2\times 5^4$.

Answer

$$\begin{aligned}\textcolor{colordef}{5}^{\textcolor{colorprop}{2}} \times \textcolor{colordef}{5}^{\textcolor{olive}{4}}&= \textcolor{colordef}{5}^{\textcolor{colorprop}{2}+\textcolor{olive}{4}} && \text{(same base, add exponents)} \\ &= \textcolor{colordef}{5}^{6}.\end{aligned}$$

Order of operations

The order of operations is a set of rules that tells us which calculations to do first in a mathematical expression.

Definition Order of Operations

To solve mathematical expressions accurately, we follow the order of operations, which is commonly remembered using the acronym PEMDAS:

P: Parentheses
E: Exponents
M: Multiplication
D: Division
A: Addition
S: Subtraction

We first do the operations at the top of the list.Multiplication and division are on the same level, so we work from left to right.Addition and subtraction are also on the same level, so we again work from left to right.

Example

Evaluate $(1+2) \times 2^3 + 4$.

Answer

$$\begin{aligned}[t](1+2) \times 2^3 + 4 &= \textcolor{colordef}{(1+2)} \times 2^3 + 4 && (\text{parentheses: } \textcolor{colordef}{(1+2)} = 3) \\ &= 3 \times \textcolor{colordef}{2^3} + 4 && (\text{exponent: } \textcolor{colordef}{2^3} = 8) \\ &= \textcolor{colordef}{3 \times 8} + 4 && (\text{multiplication: } \textcolor{colordef}{3 \times 8} = 24) \\ &= \textcolor{colordef}{24 + 4} && (\text{addition: } \textcolor{colordef}{24 + 4} = 28) \\ &= 28\end{aligned}$$

Square number	Number of grains
\(1\)	\(2\)
\(2\)	\(2 \times 2\)
\(3\)	\(2 \times 2 \times 2\)
\(\vdots\)	\(\vdots\)
\(64\)	\(\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}\)