In any math expression with more than one operation, parentheses (or brackets) are always performed first. After that, you work through exponents, then multiplication and division, and finally addition and subtraction. This hierarchy is the same worldwide, whether you learned it as PEMDAS, BODMAS, or BEDMAS.
The Four Levels of Priority
The full order breaks down into four levels, not six. Even though PEMDAS has six letters, multiplication and division share the same priority, and so do addition and subtraction. Here’s the actual hierarchy:
- Level 1: Parentheses (or brackets). Solve everything inside grouping symbols first, working from the innermost set outward.
- Level 2: Exponents. Evaluate powers and roots next.
- Level 3: Multiplication and division. Work these from left to right, in whatever order they appear.
- Level 4: Addition and subtraction. Work these from left to right, in whatever order they appear.
The critical detail most people miss is that multiplication does not outrank division, and addition does not outrank subtraction. Within each pair, you simply move left to right across the expression.
Why Left to Right Matters
The left-to-right rule is where most mistakes happen. Consider 8 ÷ 2 × 4. If you multiply first because you think multiplication comes before division, you get 8 ÷ 8 = 1. That’s wrong. Because multiplication and division sit at the same level, you resolve them left to right: 8 ÷ 2 = 4, then 4 × 4 = 16.
The same logic applies to addition and subtraction. In 2 − 3 + 4, working left to right gives you −1 + 4 = 3. If you mistakenly add first, you get 2 − 7 = −5. The left-to-right convention isn’t optional. As one Auburn University professor put it, “Everyone agrees that PEMDAS with a left to right convention is the golden rule.”
PEMDAS vs. BODMAS
Different countries use different acronyms, but the underlying rules are identical. PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is standard in the United States. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is common in the UK, Australia, and India. BEDMAS replaces “Orders” with “Exponents” and is used in Canada.
Notice that BODMAS lists division before multiplication, while PEMDAS lists multiplication before division. This trips people up, but it doesn’t matter. Both acronyms treat multiplication and division as equal. The letter order is just a consequence of fitting them into a pronounceable word.
The Implicit Multiplication Trap
One expression that routinely goes viral on social media is something like 8 ÷ 2(2+2). People split into two camps: those who get 16 and those who get 1. The disagreement comes down to whether the “2(” creates a special bond with the parentheses that should be resolved before the division.
Under strict PEMDAS rules, you solve the parentheses first (2+2 = 4), then work left to right: 8 ÷ 2 × 4 = 16. But some math conventions, particularly in higher-level algebra, treat implied multiplication (writing 2(4) instead of 2 × 4) as having higher priority than regular multiplication or division. This is called implicit multiplication, and mathematicians genuinely disagree on how to handle it.
The real lesson is that the expression is poorly written. Professional mathematicians and textbook authors avoid this kind of ambiguity by using fraction bars or extra parentheses to make their meaning clear. If you encounter one of these problems in the wild, the answer depends on which convention the writer intended, and you can’t always tell.
Another Common Mistake: Negative Signs and Exponents
What does −3² equal? Many people instinctively say 9, reasoning that negative three squared is positive nine. But standard notation treats this as −(3²), meaning the exponent applies to the 3 first, and the negative sign acts as multiplication by −1 afterward. The result is −9, not 9.
If you actually want to square negative three, you need parentheses: (−3)² = 9. Without them, the exponent binds to the number before the negation applies. This catches students off guard constantly, and it’s one of the most frequent sources of errors on standardized tests.
How It Works in Programming
If you write code, the same general idea applies, but the hierarchy is much longer. Programming languages like Java, Python, and C have over a dozen levels of operator precedence. Arithmetic follows the familiar pattern: parentheses first, then exponents (where supported), then multiplication and division, then addition and subtraction. But programming adds layers for comparison operators (like “greater than” or “equals”), logical operators (AND, OR, NOT), and assignment operators (the single equals sign that stores a value in a variable).
In most languages, arithmetic operations sit near the top of the priority list, comparisons fall in the middle, and assignment sits at the very bottom. So in an expression like x = 5 + 3 > 2, the computer adds 5 + 3 first (getting 8), then checks whether 8 is greater than 2 (true), and finally assigns that result to x. Parentheses override everything, just like in math, which is why experienced programmers use them liberally to make their intent obvious rather than relying on memorized precedence tables.
A Quick Way to Remember
If the acronyms blur together, think of it as three tiers. Grouping symbols (parentheses, brackets) always come first because they exist specifically to override the default order. Exponents come next because they represent repeated multiplication, which is a “higher-level” operation. Multiplication and division come before addition and subtraction because multiplication is repeated addition. Each tier is just a more powerful version of the one below it.
Within any single tier, go left to right. That’s the entire system. The acronyms are training wheels for this structure, and once the logic clicks, you won’t need to recite the letters anymore.