Arithmetic

Order of Operations Beyond PEMDAS

PEMDAS has six letters and describes four tiers. That mismatch is the source of nearly every mistake in this topic.

By the Gauth editorial teamHow this content is produced and checkedUpdated 10 September 2026

The idea

Precedence rules are not mathematical facts. They are typesetting conventions, chosen so that the expressions people write most often need the fewest brackets. Multiplication outranks addition because polynomials are everywhere: without the convention, $3 x^{2} + 2 x + 1$ would have to be written as $(3 \cdot (x^{2})) + (2 \cdot x) + 1$ , and nobody would tolerate that.

Once you see the rules as a compression scheme rather than a law, the four tiers make sense as a hierarchy of how strongly an operation binds to the symbols beside it. Exponents bind tightest, then multiplication and division, then addition and subtraction, with grouping symbols available to override any of it.

The reason PEMDAS misleads is that it lists six operations in a row, which invites the reading that multiplication outranks division and that addition outranks subtraction. Neither is true, and neither could be true, for a reason that is worth spelling out.

Reading like a marker means applying the convention to what is written rather than to what was meant. If you write $1/2 x$ intending $\frac{1}{2 x}$ , the convention delivers $\frac{1}{2} x$ , and the two differ by a factor of $x^{2}$ . Nobody is being pedantic when they mark that wrong: the notation has one agreed meaning, and a solution that relies on the reader guessing your intent is not a solution. The habit that prevents this is bracketing anything you would have to explain aloud.

When you need it

Order of operations is invisible when it goes right and expensive when it goes wrong. These are the situations where it does the most damage:

Substituting a negative number into a formula. Working out $b^{2} - 4 a c$ with $b = - 7$ requires $(- 7)^{2} = 49$ , and dropping the brackets silently produces $- 49$ .
Typing an expression into a calculator or spreadsheet. The machine follows the convention exactly, so an omitted bracket around a numerator produces a confidently wrong number.
Distributing over a bracket. $30 (T - 4)$ is $30 T - 120$ , and forgetting the second product is a precedence failure, not a careless one.
Evaluating a formula with a fraction bar written on one line. Anything with a multi-term numerator needs brackets that the printed version did not show.
Any expression mixing $\div$ with a bracket, which is where the genuinely ambiguous cases live.

The method

Start with the fact the acronym hides. Division is not a separate operation from multiplication; it is multiplication by a reciprocal. Likewise subtraction is addition of a negative:

a \div b = a \times \frac{1}{b}, a - b = a + (- b)

Rewrite any chain that way and the ordering question disappears, because multiplication and addition can be reordered freely. Test it on $8 \div 4 \div 2$ :

8 \times \frac{1}{4} \times \frac{1}{2} = 1

Now compare the two possible readings of the original. Left to right gives $(8 \div 4) \div 2 = 2 \div 2 = 1$ , which matches. Right to left gives $8 \div (4 \div 2) = 8 \div 2 = 4$ , which does not. The left-to-right rule is not arbitrary: it is the reading that agrees with the reciprocal form. The same argument settles subtraction, where $10 - 3 - 2$ is $10 + (- 3) + (- 2) = 5$ , matching left-to-right and contradicting the alternative.

Exponents are the exception, and for an equally concrete reason. A stacked exponent groups right to left:

2^{3^{2}} = 2^{9} = 512 not (2^{3})^{2} = 64

Grouping left to right would be pointless, because $(2^{3})^{2}$ is just $2^{6}$ and could already be written without stacking. Right-to-left grouping is the convention that lets the notation express something it otherwise could not.

The minus sign in front of a power. In $- 3^{2}$ , the negation is a multiplication by $- 1$ , so it lives on the multiplication tier — below exponents. The exponent applies to the 3 alone:

- 3^{2} = - (3^{2}) = - 9 but (- 3)^{2} = 9

This is the one place where the tools genuinely disagree, and it is worth knowing which one you are holding. Python evaluates $-3**2$ as $- 9$ , matching the mathematical convention. Excel and Google Sheets evaluate $=-3^2$ as $9$ , because their parsers give the unary minus higher precedence than the power. Neither is a bug; they are different published conventions. The practical consequence is that a formula copied from a worksheet into a spreadsheet can change value silently, so bracket the base explicitly whenever the sign matters.

Invisible grouping symbols. Parentheses are not the only ones. A fraction bar groups its numerator and its denominator separately, a radical groups everything under it, absolute value bars group their contents, and an exponent groups its own expression. That is why $\frac{7 + 5}{3 + 1}$ is $3$ , while the same characters typed on one line as $7 + 5 \div 3 + 1$ come out as $9 \frac{2}{3}$ .

The procedure, then:

Rewrite the awkward parts. Turn divisions into fractions and subtractions into added negatives whenever the expression is long enough to lose track of.
Work groupings from the inside out, remembering the invisible ones. A nested bracket is finished before its parent is touched.
Exponents, right to left. Evaluate anything sitting in an exponent before applying it.
Multiplication and division, left to right. One tier. If the division is written first, it happens first.
Addition and subtraction, left to right. One tier again.
Estimate independently. Round everything and redo it in your head. Precedence errors usually change the magnitude, not just the last digit, so a rough check catches them.

Solve a arithmetic problem now

First solution free

Three worked examples

The first exercises the two shared tiers. The second adds a bracket and a negative base. The third is the one built to punish every shortcut at once.

Worked example

12 - 4 \times 3 + 6 \div 2

1
No groupings and no exponents, so go straight to the multiplication-and-division tier. Both operations happen here, in reading order.
$4 \times 3 = 12, 6 \div 2 = 3$
2
Substitute both results. The expression is now purely additive.
$12 - 12 + 3$
3
Work left to right. Doing the addition first, as PEMDAS's letter order suggests, would give 12 − 15 = −3, which is wrong.
$(12 - 12) + 3 = 0 + 3 = 3$
4
Check by rewriting as added negatives: 12 + (−12) + 3, which can be summed in any order and still gives 3.
$12 + (- 12) + 3 = 3$

Answer

3

Worked example

8 - 3 (4 - 6)^{2} \div 2

1
Innermost grouping first. The bracket is a single grouping, so evaluate the subtraction inside it before anything else touches it.
$4 - 6 = - 2$
2
Now the exponent. The base is the whole bracketed value, including its sign, because the brackets are what the exponent attaches to.
$(- 2)^{2} = 4$
3
The expression is now 8 − 3 × 4 ÷ 2. Multiplication and division share a tier, so work left to right: the multiplication is written first.
$3 \times 4 = 12$
4
Then the division.
$12 \div 2 = 6$
5
Finally the subtraction. Note that the 3 was never subtracted from the 8 — it belongs to the product.
$8 - 6 = 2$

Answer

2

Worked example

\frac{- 3 ^{2} + ( - 4 ) ^{2}}{5 - 2 \cdot 3 ^{0}} - 6 \div 3 \cdot 2

1
The fraction bar is a grouping symbol, so the numerator and the denominator are each finished before the division between them happens. Start with the numerator, where the two minus signs behave differently.
$- 3^{2} = - (3^{2}) = - 9, (- 4)^{2} = 16$
2
Add them. The first term is negative because the exponent applied only to the 3; the second is positive because the brackets put the sign inside.
$- 9 + 16 = 7$
3
Now the denominator. Any non-zero number raised to the power zero is 1, and that exponent is evaluated before the multiplication by 2.
$3^{0} = 1 ⟹ 2 \cdot 1 = 2 ⟹ 5 - 2 = 3$
4
The fraction is therefore seven thirds. Do not convert to a decimal; the final answer stays exact.
$\frac{7}{3}$
5
Handle the trailing term separately. Division and multiplication share a tier, so the division is done first because it is written first. Reading it as 6 ÷ (3 · 2) would give 1, which is the trap.
$6 \div 3 \cdot 2 = 2 \cdot 2 = 4$
6
Subtract, using a common denominator of 3.
$\frac{7}{3} - 4 = \frac{7}{3} - \frac{12}{3} = - \frac{5}{3}$
7
Estimate to check: the fraction is a bit over 2, minus 4, so an answer near −1.7 is expected. −5/3 is −1.67.
$- \frac{5}{3} \approx - 1.67$

Answer

- \frac{5}{3}

Find the planted mistake

One line below breaks the rules. The final answer looks tidy enough to pass a quick glance, which is exactly how this error survives into marked work.

Evaluate 20 − 6 ÷ 2 × 5 + 3²

One of these lines is wrong. Click it.

Where people go wrong

Doing all multiplication before any division

$6 \div 3 \times 2$ is $4$ , not $1$ . The M appearing before the D in the acronym is an artefact of needing a pronounceable word, not a statement about precedence. The same applies to addition and subtraction: $10 - 3 + 2$ is $9$ , not $5$ .

Losing the brackets around a negative base

When you substitute $x = - 4$ into $x^{2}$ , write $(- 4)^{2}$ . Writing $- 4^{2}$ asks a different question and answers $- 16$ . This single habit accounts for a large share of the discriminant errors in quadratic formula problems, where $b$ is negative more often than not.

Treating a fraction bar as ordinary division

Retyping $\frac{- b + d}{2 a}$ onto one line as $- b + d \div 2 a$ divides only the radical, and then the $a$ multiplies rather than divides. The numerator and denominator each need their own brackets: $(-b + sqrt(d))/(2*a)$ .

Arguing about deliberately ambiguous expressions

$8 \div 2 (2 + 2)$ has no defensible single answer. Treating implicit multiplication like explicit multiplication gives 16, and giving juxtaposition higher precedence — standard in a lot of scientific typesetting — gives 1. Both camps are applying a real convention consistently. The mathematically useful response is not to pick a side but to write $\frac{8}{2 ( 2 + 2 )}$ or $\frac{8}{2} (2 + 2)$ and remove the question.

Practice

Evaluate each exactly. Where the answer is a fraction, write it as $a/b$ in lowest terms.

Four expressions, four different traps

Answers are checked here — nothing is sent anywhere.

1
$7 + 3 \times 4^{2} \div 8$
2
$- 5^{2} + (- 5)^{2}$
3
$24 \div 4 \div 3$
4
$2 + 3 [8 - 2 (5 - 3)]$

Where this goes next

The exponent tier is the one with the most hidden structure, and the rules for combining and splitting powers and radicals explain why $3^{0}$ had to be 1 in the third example rather than 0. Precedence errors also cluster in a handful of recognisable places, catalogued with fixes in the twelve algebra mistakes that quietly cost marks. When an expression is long enough that a slip is likely, the algebra solver evaluates it tier by tier so you can see exactly which step your working diverged at.

Frequently asked questions

Is it PEMDAS or BODMAS, and do they disagree?

They are the same rule with different words. P and B both mean grouping symbols, E and O both mean exponents or orders, and the remaining four letters are identical. Neither acronym disagrees with the other. Both mislead in the same way, by listing six letters when there are only four tiers.

Why is division done at the same level as multiplication?

Because dividing by b is the same operation as multiplying by 1/b. Once you rewrite every division that way, the expression is a chain of multiplications, and the order within a chain of multiplications cannot change the result. Left-to-right is simply the reading convention that makes the un-rewritten form agree with the rewritten one.

Why is -3^2 equal to -9?

Because the exponent binds tighter than the negation. The expression means -(3^2), which is -9. Writing (-3)^2 puts the minus sign inside the grouping, so the whole of -3 gets squared and the answer is 9. This is not a quirk of calculators; it is the same precedence that lets you write -x^2 for a downward parabola without brackets.

What is the correct answer to 8 ÷ 2(2+2)?

The expression is genuinely ambiguous and the honest answer is that it should not be written that way. Most calculators and programming languages treat implicit multiplication at the same precedence as explicit multiplication, giving 16. A substantial body of scientific typesetting gives juxtaposition higher precedence, giving 1. Neither convention is wrong; the notation is. Use a fraction bar and the ambiguity disappears.

Does the order of operations apply inside a fraction?

Yes, and the fraction bar is itself a grouping symbol. Everything above the bar is evaluated as a unit and everything below it as another unit, then the division happens last. That is why (7 + 5)/(3 + 1) needs brackets when written on one line as 7 + 5 ÷ 3 + 1 would mean something completely different.

Keep going

Algebra solverEvaluates and solves with each operation shown in the order it was applied.Exponent and radical rulesWhy anything to the power zero is 1, derived rather than stated.The quadratic formulaWhere a missing bracket around a negative b does the most damage.12 algebra mistakes that cost marksEach one shown wrong, then right, with the misconception named.