BODMAS, Simplification & Approximation

Order of operations — BODMAS/VBODMAS rule, types of brackets and their order of removal, vinculum (bar); simplification using algebraic identities (square and cube identities, difference of squares, a³ + b³ + c³ − 3abc); multiplication shortcuts (numbers near bases, multiplying by 5, 25, 99, 101, 11); simplification of expressions with fractions, decimals, percentages and square roots; approximation techniques — rounding, percentage approximation, square root and division approximation, choosing closest options; digit-sum (casting out nines) checks — with fully worked examples.

BODMAS, Simplification & Approximation

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📑 Contents (7 sections)

Last reviewed 16 Sept 2026 · 6 min read

Order of operations — BODMAS / VBODMAS

Letter	Operation	Order
V	Vinculum (bar over numbers — e.g. $\overline{8 - 3}$ )	First
B	Brackets — in order: ( ) → { } → [ ] (innermost first)	Second
O	Of (means multiplication, e.g. ½ of 20) and orders/powers/roots	Third
D	Division	Fourth (D and M from left to right)
M	Multiplication
A	Addition	Last (A and S from left to right)
S	Subtraction

Key point: Division and multiplication have equal priority — evaluate left to right; similarly addition and subtraction.

Algebraic identities for simplification

FormulaUseful identities

$(a + b)^{2} = a^{2} + 2 ab + b^{2}$ ; $(a - b)^{2} = a^{2} - 2 ab + b^{2}$
$a^{2} - b^{2} = (a + b) (a - b)$
$(a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})$ ; $(a + b)^{2} - (a - b)^{2} = 4 ab$
$(a + b)^{3} = a^{3} + b^{3} + 3 ab (a + b)$ ; $(a - b)^{3} = a^{3} - b^{3} - 3 ab (a - b)$
$a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ ; $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$
$a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)$ ; if $a + b + c = 0$ , then $a^{3} + b^{3} + c^{3} = 3 ab c$
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)$

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