The PEMDAS rule tells us the correct order for evaluating arithmetic expressions.
It prevents ambiguity. Without a fixed order, the same expression could produce different answers depending on which operation someone performs first.
1. What PEMDAS means
PEMDAS stands for:
| Letter |
Meaning |
What to do |
| P |
Parentheses |
Evaluate expressions inside parentheses, brackets, or braces first. |
| E |
Exponents |
Evaluate powers after grouping symbols. |
| M/D |
Multiplication and division |
Evaluate multiplication and division from left to right. |
| A/S |
Addition and subtraction |
Evaluate addition and subtraction from left to right. |
The same rule is also called BODMAS in some countries:
brackets, orders, division/multiplication, addition/subtraction.
2. The most important correction
PEMDAS does not mean multiplication always happens before division.
Multiplication and division have the same priority, so they are done from left to right.
\[
\begin{aligned}
24\div3\times2
&= 8\times2\\
&= 16.
\end{aligned}
\]
It would be wrong to force the multiplication first unless parentheses tell you to do so:
\[
\begin{aligned}
24\div(3\times2)
&=24\div6\\
&=4.
\end{aligned}
\]
These are different expressions because the second one has parentheses.
3. Addition and subtraction also go left to right
Addition and subtraction also share the same priority.
\[
\begin{aligned}
10-3+2
&=7+2\\
&=9.
\end{aligned}
\]
It is not correct to do \(3+2\) first unless the expression is written as
\[
\begin{aligned}
10-(3+2).
\end{aligned}
\]
4. Parentheses, brackets, and braces
Grouping symbols tell us to evaluate an expression as one unit.
The common grouping symbols are
\[
\begin{aligned}
( \ ),\qquad [ \ ],\qquad \{ \}.
\end{aligned}
\]
For example:
\[
\begin{aligned}
2(3+4)
&=2(7)\\
&=14.
\end{aligned}
\]
Without the parentheses, \(2\times3+4\) would be
\[
\begin{aligned}
2\times3+4
&=6+4\\
&=10.
\end{aligned}
\]
Parentheses can completely change the value of an expression.
5. Exponents
Exponents are evaluated after grouping symbols and before multiplication/division.
\[
\begin{aligned}
3+4\times2^2
&=3+4\times4\\
&=3+16\\
&=19.
\end{aligned}
\]
The power \(2^2\) is evaluated before the multiplication \(4\times4\).
6. Full worked example
Consider the expression:
\[
\begin{aligned}
8+2\times(3+4)^2\div7.
\end{aligned}
\]
First evaluate the parentheses:
\[
\begin{aligned}
3+4 &= 7.
\end{aligned}
\]
The expression becomes:
\[
\begin{aligned}
8+2\times7^2\div7.
\end{aligned}
\]
Next evaluate the exponent:
\[
\begin{aligned}
7^2 &= 49.
\end{aligned}
\]
The expression becomes:
\[
\begin{aligned}
8+2\times49\div7.
\end{aligned}
\]
Now multiply and divide from left to right:
\[
\begin{aligned}
2\times49 &= 98,\\
98\div7 &= 14.
\end{aligned}
\]
Finally add:
\[
\begin{aligned}
8+14 &= 22.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{8+2\times(3+4)^2\div7=22}.
\end{aligned}
\]
7. Why the result is not 36
A common mistake is to group the expression incorrectly after the exponent.
Once \((3+4)^2\) becomes \(49\), the expression is
\[
\begin{aligned}
8+2\times49\div7.
\end{aligned}
\]
Since multiplication and division are on the same level, work left to right:
\[
\begin{aligned}
2\times49\div7
&=98\div7\\
&=14.
\end{aligned}
\]
So the final result is \(8+14=22\).
8. Nested grouping
If grouping symbols are nested, evaluate the innermost group first.
\[
\begin{aligned}
\{12-[2+3]\}\times4
&=\{12-5\}\times4\\
&=7\times4\\
&=28.
\end{aligned}
\]
Brackets and braces are not a new type of operation. They are just grouping symbols.
9. Exponent direction
In standard mathematical convention, repeated exponents are usually evaluated from the top down:
\[
\begin{aligned}
2^{3^2}
&=2^9\\
&=512.
\end{aligned}
\]
If you want a different grouping, use parentheses:
\[
\begin{aligned}
(2^3)^2
&=8^2\\
&=64.
\end{aligned}
\]
10. Formula summary
| Priority |
Operation |
Direction |
Example |
| 1 |
Parentheses, brackets, braces |
Inside first |
\((3+4)=7\) |
| 2 |
Exponents |
Use grouping when needed |
\(7^2=49\) |
| 3 |
Multiplication and division |
Left to right |
\(24\div3\times2=16\) |
| 4 |
Addition and subtraction |
Left to right |
\(10-3+2=9\) |
11. Common mistakes
- Doing all operations from left to right without checking priority.
- Multiplying before dividing even when division comes first.
- Adding before subtracting even when subtraction comes first.
- Ignoring parentheses or brackets.
- Forgetting that exponents happen before multiplication and division.
- Forgetting to use parentheses when a different grouping is intended.
Key idea: PEMDAS is not a list of separate left-to-right commands. It is a priority system, with multiplication/division tied together and addition/subtraction tied together.
