Order of Operations with Integers

You already know PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) from working with whole numbers. The order of operations does not change when negative numbers enter the picture. What does change is the number of places where a sign error can sneak in. A misplaced negative sign or a forgotten parenthesis can flip an answer from positive to negative or vice versa.

This page focuses on applying PEMDAS correctly when expressions involve negative integers, with special attention to the traps that catch most learners.

PEMDAS Review

The order of operations tells you which calculations to do first:

1. P — Parentheses (and other grouping symbols)
2. E — Exponents
3. M/D — Multiplication and Division (left to right)
4. A/S — Addition and Subtraction (left to right)

Within each level, work left to right.

The Most Common Sign Traps

Before diving into examples, here are the errors to watch for:

Trap 1: $-a^2$ vs. $(-a)^2$

$-3^2 = -(3^2) = -9$

$(-3)^2 = (-3) \times (-3) = 9$

Without parentheses around the negative sign, the exponent applies only to the $3$. The negative sign is a separate operation applied after the exponent.

Trap 2: Subtracting a negative

$5 - (-3) = 5 + 3 = 8$

When you see “minus a negative,” rewrite it as addition. Two negatives next to each other become a positive.

Trap 3: Distributing a negative sign

$-(4 + 7) = -4 - 7 = -11$

The negative sign outside the parentheses applies to every term inside.

Worked Examples

Example 1: Basic PEMDAS with Negatives

Evaluate: $\;(-3) + 4 \times (-2)$

Step 1: No parentheses to simplify (the parentheses here just clarify signs). No exponents.

Step 2: Multiplication first: $4 \times (-2) = -8$.

Step 3: Addition: $(-3) + (-8) = -11$.

$(-3) + 4 \times (-2) = -11$

Example 2: Parentheses Change Everything

Evaluate: $\;((-3) + 4) \times (-2)$

Step 1: Parentheses first: $(-3) + 4 = 1$.

Step 2: Multiply: $1 \times (-2) = -2$.

$((-3) + 4) \times (-2) = -2$

Notice how adding parentheses changed the answer from $-11$ to $-2$.

Example 3: Exponents with Negative Bases

Evaluate: $\;(-2)^3 - 5$

Step 1: Exponent first: $(-2)^3 = (-2) \times (-2) \times (-2) = -8$.

Three negative factors (odd count), so the result is negative.

Step 2: Subtract: $-8 - 5 = -13$.

$(-2)^3 - 5 = -13$

Example 4: The $-a^2$ Trap

Evaluate: $\;-4^2 + 10$

Step 1: The exponent applies only to $4$, not to the negative sign: $4^2 = 16$.

Step 2: Apply the negative: $-16$.

Step 3: Add: $-16 + 10 = -6$.

$-4^2 + 10 = -6$

If the problem were $(-4)^2 + 10$, the answer would be $16 + 10 = 26$. Parentheses make all the difference.

Example 5: Multi-Step Expression

Evaluate: $\;(-6) \div 2 + (-3)^2 \times 4 - 1$

Step 1: Exponents first: $(-3)^2 = 9$.

Step 2: Multiplication and division, left to right:

$(-6) \div 2 = -3$

$9 \times 4 = 36$

Step 3: Addition and subtraction, left to right:

$-3 + 36 - 1 = 33 - 1 = 32$

$(-6) \div 2 + (-3)^2 \times 4 - 1 = 32$

More Practice Scenarios

Nested Parentheses

Evaluate: $\;-2 \times (3 - (1 - 5))$

Step 1: Innermost parentheses: $1 - 5 = -4$.

Step 2: Outer parentheses: $3 - (-4) = 3 + 4 = 7$.

Step 3: Multiply: $-2 \times 7 = -14$.

$-2 \times (3 - (1 - 5)) = -14$

Fraction Bar as Grouping

A fraction bar acts as a grouping symbol, meaning you evaluate the numerator and denominator separately before dividing.

Evaluate: $\;\dfrac{(-4) + 8}{-6 + 4}$

Numerator: $(-4) + 8 = 4$

Denominator: $-6 + 4 = -2$

Divide: $\dfrac{4}{-2} = -2$

$\frac{(-4) + 8}{-6 + 4} = -2$

Practice Problems

Problem 1: Evaluate $\;5 + (-3) \times 2$

Problem 2: Evaluate $\;(-5)^2 - 3 \times 4$

Problem 3: Evaluate $\;-2^3 + 6 \div (-3)$

Problem 4: Evaluate $\;(10 - 14) \times (-3) + 2$

Problem 5: Evaluate $\;\dfrac{(-2)^3 + 4}{-8 \div 2}$

Key Takeaways

• PEMDAS does not change when negative numbers are involved. The order remains: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Watch the exponent trap: $-3^2 = -9$ but $(-3)^2 = 9$. Parentheses determine whether the negative is part of the base.
• Subtracting a negative becomes adding a positive: $a - (-b) = a + b$.
• Fraction bars act as grouping symbols. Evaluate the top and bottom separately before dividing.
• When in doubt, add extra parentheses to clarify what each sign applies to. It is better to over-clarify than to make a sign error.

Return to Arithmetic for more foundational math topics.