You know the order of operations. Parentheses let you change that order: whatever is inside them is worked first. When an expression needs groups inside groups, we use bigger symbols so the eye can match the pairs: parentheses $(\;)$, brackets $[\;]$ and braces $\{\;\}$.
Key idea
Start with the innermost grouping symbols and work outward. Inside each pair, use the usual order: multiply and divide, then add and subtract.
Why it works
A grouping symbol says “treat this as one number.” In $3 \times (8 + 2)$ you multiply 3 by the amount $8 + 2$, which is 10, so the value is 30. Without the parentheses, $3 \times 8 + 2$ is 26. Brackets and braces mean exactly the same as parentheses; they only make nested groups easier to read.
Worked examples
Worked example: Brackets around parentheses
Find $2 \times [5 + (6 - 2)]$.
- The innermost group is $(6 - 2) = 4$. Now: $2 \times [5 + 4]$.
- Next, the brackets: $5 + 4 = 9$. Now: $2 \times 9$.
- Multiply: $2 \times 9 = 18$.
Answer: $18$
Worked example: Braces too
Find $\{20 - [3 \times (4 + 1)]\} \div 5$.
- Innermost: $(4 + 1) = 5$. Now: $\{20 - [3 \times 5]\} \div 5$.
- Brackets: $3 \times 5 = 15$. Now: $\{20 - 15\} \div 5$.
- Braces: $20 - 15 = 5$. Now: $5 \div 5$.
- Divide: $5 \div 5 = 1$.
Answer: $1$
| Expression | First step | Value |
|---|---|---|
| $3 \times (8 + 2)$ | $8 + 2 = 10$ | $30$ |
| $3 \times 8 + 2$ | $3 \times 8 = 24$ | $26$ |
| $(3 \times 8) + 2$ | $3 \times 8 = 24$ | $26$ |
Common mistake
Working the outer symbols first, or dropping the inner ones. In $2 \times [5 + (6 - 2)]$, doing $2 \times 5$ first gives the wrong value. Check: find the pair of symbols with nothing grouped inside it. That pair is always first.