BODMAS and Nested Brackets

Learning outcome

Evaluate expressions in a consistent order, handle nested brackets and explain how parentheses change the meaning of a squared negative number.

Concepts and reasons

An expression combines numbers and operations. BODMAS is a shared convention for reading it, not six separate priority levels. It prevents different readers from assigning different meanings to the same expression.

B: Brackets. Evaluate the innermost group first, then work outward. Round brackets (), square brackets [] and braces {} all group expressions. Their nesting, not their shape, determines which group comes first. Apply the usual operation order inside each group.

O: Orders. Calculate powers after their bases have been determined. Here exponents are positive integers: squaring means multiplying a base by itself, not multiplying it by the exponent.

D and M: Division and multiplication have equal precedence. Work from left to right. BODMAS does not mean that every division must be completed before every multiplication.

A and S: Addition and subtraction have equal precedence. Work from left to right. It does not mean doing all additions before subtractions. Complete multiplication and division before starting this level.

A unary negative sign applies to one expression; subtraction operates between two expressions. Parentheses can make a negative number the base of a power. Without them, a power is evaluated before a leading unary minus: -5^2 means -(5^2), whereas (-5)^2 squares the entire negative base.

This difference is about scope: which part does the sign affect? Likewise, brackets following a division sign specify the whole divisor. Always write × explicitly when multiplication is intended; do not rely on adjacent numbers and brackets.

Worked examples

Example 1 — Equal precedence. Evaluate 36 ÷ 6 × 3 - 8 + 2. First, 36 ÷ 6 = 6, then 6 × 3 = 18. Now work left to right: 18 - 8 = 10 and 10 + 2 = 12. Do not combine 6 × 3 into a single divisor.

Example 2 — Nested brackets. Evaluate 72 ÷ {3 × [8 - (5 - 3)]}. The inner difference is 2. The square bracket becomes 8 - 2 = 6; the brace becomes 3 × 6 = 18. Therefore 72 ÷ 18 = 4.

Example 3 — Signs and squares. Evaluate -5^2 + (-5)^2 - 2 × [3 - (-4)]. Here -5^2 = -25, but (-5)^2 = 25. The bracket equals 7, so the expression becomes -25 + 25 - 14 = -14. The squared terms cancel; the final subtraction remains.

Common mistakes

Do not calculate every operation from left to right regardless of priority. Do not give division priority over multiplication, or addition over subtraction. A divisor that simplifies to zero makes the expression undefined.

Practice questions

1. Evaluate 64 ÷ 8 × 3 - 7 + 2.
2. Evaluate 90 ÷ {5 × [7 - (4 + 1)]}.
3. Evaluate -6^2 + (-6)^2 + 18 ÷ (-3).
4. Is 20 ÷ [3 × (5 - 3) - 6] defined? Explain.

Answers and explanations

1. 19. Division and multiplication give 64 ÷ 8 = 8, then 8 × 3 = 24. Finally, 24 - 7 + 2 = 17 + 2 = 19.
2. 9. First 4 + 1 = 5, then 7 - 5 = 2. The divisor becomes 5 × 2 = 10, so 90 ÷ 10 = 9.
3. -6. The terms are -36, 36 and -6 respectively. Thus -36 + 36 - 6 = -6. Parentheses determine whether the minus sign is squared.
4. Undefined. The divisor is 3 × 2 - 6 = 0. The expression requires 20 ÷ 0, which is not permitted; it does not equal zero.