\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Exponent

Exponents are an efficient way of expressing repeated multiplication, and 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?

Square number Grain number
\(1\) \(2\)
\(2\) \(2 \times 2\)
\(3\) \(2 \times 2 \times 2\)
\(\vdots\) \(\vdots\)
\(64\) \(\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}\)
Rather than writing \(\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}\), we can write this product as \(2^{64}\). Using a calculator, \(2^{64}=18\,446\,744\,073\,709\,551\,616\) which is over 2,000 times the annual world production of wheat, which in the period 2020-21 was an estimated 800 million metric tonnes.


Definition Exponentiation
Exponentiation is repeated multiplication of a number by itself:
Example
Write using exponent notation: \(5 \times 5 \times 5\)

\(5 \times 5 \times 5 = 5^3\)

Definition Vocabulary
$$\begin{array}{|c|c|c|c|}\hline\text{Value} & \text{Expanded form} & \text{Exponent notation} & \text{Spoken form} \\\hline2 & 2 & 2^1 & 2\ \text{or}\ 2\ \text{raised to the power}\ 1 \\4 & 2 \times 2 & 2^2 & 2\ \text{squared or}\ 2\ \text{raised to the power}\ 2 \\8 & 2 \times 2 \times 2 & 2^3 & 2\ \text{cubed or}\ 2\ \text{raised to the power}\ 3 \\16 & 2 \times 2 \times 2 \times 2 & 2^4 & 2\ \text{raised to the power}\ 4 \\32 & 2 \times 2 \times 2 \times 2 \times 2 & 2^5 & 2\ \text{raised to the power}\ 5 \\\hline\end{array}$$
Example
Find the value for \(2^3\).

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

Order of operations

The order of operations is a set of guidelines that help us solve mathematical expressions in a consistent manner.
Definition Order of Operations
To solve mathematical expressions accurately, we follow the order of operations, which is commonly remembered using the acronym PEMDAS:
  1. P: Parentheses
  2. E: Exponents
  3. M: Multiplication
  4. D: Division
  5. A: Addition
  6. S: Subtraction
The order of operations proceeds from top to bottom, meaning we start with parentheses, then exponents, and so on. However, multiplication and division, as well as addition and subtraction, have the same level of priority. In these cases, we work from left to right.
Example
Evaluate \((1+2) \times 2^3 + 4\)

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