When an expression mixes operations, the order you carry them out changes the answer. So mathematicians everywhere agreed on one order — learn it once and every calculation comes out the same.
Why we need an agreed order
Take \( 2 + 3 \times 4 \). Add first and you get \( 20 \); multiply first and you get \( 14 \). They can't both be right, so there must be a rule. The rule says multiplication comes first, so the answer is \( 14 \).
Without a shared convention, the same string of symbols would mean different things to different people — and a calculator in Tokyo would disagree with one in London. The order of operations exists precisely so that an expression has one meaning, no matter who reads it.
PEMDAS, step by step
The agreed order is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. (In Britain it's BODMAS — Brackets, Orders, Division/Multiplication, Addition/Subtraction — the very same rule.)
The crucial subtlety lives inside that list. Multiplication and division share one rank and are done left to right; addition and subtraction share the next rank, also left to right. Neither pair has a winner — you simply work along the line as you meet them.
The "E" stands for exponents — repeated multiplication, like \( 3^2 = 3 \times 3 \). Exponents are resolved early, right after parentheses and before any multiplication. Drag the slider below to see how a small base grows once an exponent is applied.
A full walk-through
Let's evaluate one expression slowly, doing exactly one thing per line so nothing slips past:
\[ 4 + 2 \times 3^2 - (5 - 2) \]Start with the parentheses. The bracket \( (5 - 2) = 3 \), so the expression becomes:
\[ 4 + 2 \times 3^2 - 3 \]Next the exponent: \( 3^2 = 9 \), which gives:
\[ 4 + 2 \times 9 - 3 \]Now the multiplication: \( 2 \times 9 = 18 \):
\[ 4 + 18 - 3 \]Finally add and subtract left to right: \( 4 + 18 = 22 \), then \( 22 - 3 \):
\[ 22 - 3 = 19. \]So the value is \( 19 \). Notice how each line is the whole expression rewritten — never two changes squeezed into one.
The traps that catch everyone
Almost every mistake comes from one misreading: thinking multiplication always beats division, or addition always beats subtraction. It doesn't. Equal-rank operations go left to right, in the order they appear.
Watch what happens when you ignore that:
\[ 8 \div 2 \times 2 = 4 \times 2 = 8 \]The division comes first only because it's further left — the answer is \( 8 \), not \( 2 \). The same trap appears with plus and minus:
\[ 10 - 4 + 1 = 6 + 1 = 7 \]Here the subtraction and addition happen in order, giving \( 7 \), not \( 5 \). A useful way to think about it: a minus sign belongs to the number right after it, so \( 10 - 4 + 1 \) is really \( 10 + (-4) + 1 \), and the order no longer matters.
Equal ranks M and D are equal in rank — do them left to right. The same goes for A and S. PEMDAS has SIX letters but only FOUR levels.
- There are no parentheses or exponents, so start with division and multiplication, working left to right.
- The leftmost is \( 12 \div 3 = 4 \).
- Then \( 4 \times 2 = 8 \).
- Finally the addition: \( 6 + 8 \).
So \( 6 + 12 \div 3 \times 2 = \mathbf{14} \). The division ran before the multiplication only because it came first.
- Parentheses first: \( (3 + 2) = 5 \).
- Then the exponent: \( 5^2 = 25 \).
- Now the multiplication: \( 4 \times 3 = 12 \).
- Finally the subtraction: \( 25 - 12 \).
So \( (3 + 2)^2 - 4 \times 3 = \mathbf{13} \).
Tip Rewrite the whole expression on a new line after each single step. Never do two operations at once — that's where mistakes hide.
Practice
Try each one yourself, then reveal the full solution.
1. Compute \( 5 + 2 \times 3 \).
Multiplication outranks addition, so do it first: \( 2 \times 3 = 6 \).
Then add: \( 5 + 6 \).
So \( 5 + 2 \times 3 = \mathbf{11} \).
2. Compute \( (8 - 3)^2 \div 5 \).
Parentheses first: \( (8 - 3) = 5 \).
Then the exponent: \( 5^2 = 25 \).
Finally the division: \( 25 \div 5 \).
So \( (8 - 3)^2 \div 5 = \mathbf{5} \).
3. Compute \( 20 - 6 \div 2 + 1 \).
Division comes before addition and subtraction: \( 6 \div 2 = 3 \).
That leaves \( 20 - 3 + 1 \). Now work left to right: \( 20 - 3 = 17 \), then \( 17 + 1 \).
So \( 20 - 6 \div 2 + 1 = \mathbf{18} \).