Math Core

When an expression has more than one operation, everyone needs to agree on which one to do first. Otherwise $3 + 4 \times 2$ could mean $14$ or $11$ . Mathematicians settled this long ago with the order of operations.

The order

Order of operations

Grouping symbols first: parentheses $(\,)$ , brackets $[\,]$ , fraction bars, and absolute value bars. Work from the innermost group outward.
Exponents (powers and roots).
Multiplication and division, from left to right.
Addition and subtraction, from left to right.

Many students remember this as PEMDAS ("Please Excuse My Dear Aunt Sally"). The acronym hides something important, though: multiplication and division are the same level, and so are addition and subtraction. You do whichever comes first reading left to right.

Common mistake

In $12 \div 3 \times 2$ , division comes first because it is on the left: $12 \div 3 \times 2 = 4 \times 2 = 8$ . It is not $12 \div 6 = 2$ .

Worked example: Grouping, then exponents

Evaluate $2 + 3(5 - 1)^2$ .

\begin{aligned} 2 + 3(5 - 1)^2 &= 2 + 3(4)^2 && \text{parentheses} \\ &= 2 + 3 \cdot 16 && \text{exponent} \\ &= 2 + 48 && \text{multiply} \\ &= 50 && \text{add} \end{aligned}

Worked example: A fraction bar is a grouping symbol

Evaluate $\dfrac{18 - 2 \cdot 3}{4 + 2}$ .

Simplify the top and the bottom separately first. Top: $18 - 6 = 12$ . Bottom: $4 + 2 = 6$ . So the value is $\dfrac{12}{6} = 2$ .

Negative signs and exponents

An exponent applies only to what is directly in front of it.

$(-3)^2 = (-3)(-3) = 9$ , because the parentheses put the negative inside the base.
$-3^2 = -(3 \cdot 3) = -9$ , because the exponent applies only to the $3$ .

Tip

When you're unsure, rewrite $-3^2$ as $-1 \cdot 3^2$ . The exponent happens before the multiplication.