Math Core

You already know the order of operations from earlier grades. Now the numbers can be negative, and that adds two new traps: deciding what an exponent applies to, and keeping track of signs through several steps. This lesson puts everything from the unit together.

The order, one more time

Order of operations

Grouping symbols: parentheses $(\,)$ , brackets $[\,]$ , absolute value bars $|\,|$ , and fraction bars. Work from the inside out.
Exponents.
Multiplication and division, left to right.
Addition and subtraction, left to right.

Multiplication and division are one level; so are addition and subtraction. At each level, work left to right. For example, $-20 \div 5 \times (-2)$ means $(-4) \times (-2) = 8$ , not $-20 \div (-10)$ .

Negative signs and exponents

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

In $(-4)^2$ , the base is $-4$ : $(-4)(-4) = 16$ .
In $-4^2$ , the base is just $4$ . The negative sign is applied after the exponent: $-(4 \cdot 4) = -16$ .

Common mistake

$-4^2$ is not $16$ . Read it as "the opposite of $4^2$ ." If you want the negative number squared, the parentheses must be written: $(-4)^2$ .

The same idea applies when you substitute a negative number into an expression. If $x = -3$ , then $x^2$ means $(-3)^2 = 9$ . Always put parentheses around a negative number when you substitute it.

Subtraction signs versus negative signs

In $7 - -2$ , the first dash means subtract and the second means negative. Mathematicians write it as $7 - (-2)$ to make that clear, and it equals $7 + 2 = 9$ . As you simplify, keep asking: is this dash an operation or part of a number?

Worked example: Signs through the steps

Evaluate $-6 + 3(-4)$ .

\begin{aligned} -6 + 3(-4) &= -6 + (-12) && \text{multiply first} \\ &= -18 && \text{add} \end{aligned}

Worked example: Exponents with negatives

Evaluate $-2^4 + (-2)^3$ .

\begin{aligned} -2^4 + (-2)^3 &= -16 + (-8) && \text{exponents: } -(2^4) = -16,\ (-2)^3 = -8 \\ &= -24 && \text{add} \end{aligned}

Worked example: Nested grouping

Evaluate $5 - 2[\,3 - (1 - 6)\,]$ .

\begin{aligned} 5 - 2[\,3 - (1 - 6)\,] &= 5 - 2[\,3 - (-5)\,] && \text{innermost parentheses} \\ &= 5 - 2[\,8\,] && 3 - (-5) = 3 + 5 \\ &= 5 - 16 && \text{multiply} \\ &= -11 && \text{subtract} \end{aligned}

Worked example: Fraction bar and substitution

Evaluate $\dfrac{x^2 - 10}{x + 1}$ when $x = -4$ .

Substitute with parentheses: $\dfrac{(-4)^2 - 10}{(-4) + 1}$ .

Top: $16 - 10 = 6$ . Bottom: $-4 + 1 = -3$ . So the value is $\dfrac{6}{-3} = -2$ .

Tip

After each step, rewrite the whole expression, not just the part you changed. Most sign errors come from losing a negative sign that was sitting off to the side.