DMAS — Order of
Operations
Master deep nested brackets, powers of negatives, complex fractions and multi-step word problems.
Core DMAS — Advanced Review
At this level you must apply DMAS flawlessly across deeply nested brackets, fractional bases, negative powers, and multi-step word problems. The rules never change — only the complexity increases.
Brackets first (innermost to outermost). Then powers. Then × and ÷ left to right. Finally + and − left to right. Every rule applies at every level of nesting.
Without brackets the power applies first, then the negative. With brackets the negative is included inside the power. This is the most common advanced mistake.
When × and ÷ have equal priority, always work left to right. 24÷6=4 first, then 4×2=8. Never do 6×2=12 first.
Deep Nested Brackets
When brackets appear inside brackets, always resolve from the innermost outward. Use different bracket types — ( ), [ ], { } — as visual guides, but the rule is always the same: innermost first.
Negative Numbers & Powers
Powers of negative numbers follow strict rules. The bracket determines whether the negative is part of the base.
A negative base raised to an even power is always positive: neg × neg = pos, and this repeats for every pair.
A negative base raised to an odd power stays negative: the final multiplication is always neg × pos = neg.
Without brackets, the power applies to the number only, then the negative sign is applied. So −2³ = −8, not +8.
Fractions in Expressions
Fractions follow the same DMAS rules. A fraction multiplied or divided behaves like any other factor.
Dividing by ½ doubles the number. Dividing by ⅓ triples it. Always apply this before + and −.
Resolve the bracket first (8+4=12), then multiply by the fraction. Never distribute prematurely.
Word Problems — Translating to DMAS
Word problems require you to build the expression first, then apply DMAS. Key phrases to look for:
Fractions and percentages of a quantity always mean multiplication.
Actions that happen in sequence must be wrapped in brackets to preserve order.
Distribution problems use division; accumulation problems use multiplication.
Practice Exams
Three advanced exams — each 20 questions. Use 💡 hints on any question you find challenging.