Algebra
Order of Operations Explained
The order of operations is a parsing convention. Grouping symbols come first, followed by exponents, multiplication and division from left to right, then addition and subtraction from left to right.
Reviewed
Worked example 1
Evaluate 3 + 4 x 2^3
Powers precede multiplication, which precedes addition.
2^3 = 84 x 8 = 323 + 32 = 35
Result: 35
Trace one expression one operation at a time
For 3 + 4 x 2^3, each row shows the complete expression after exactly one priority level has been resolved.
| Stage | Expression | Reason |
|---|---|---|
| Start | 3 + 4 x 2^3 | The exponent has highest priority |
| Exponent | 3 + 4 x 8 | 2^3 = 8 |
| Multiplication | 3 + 32 | Multiplication precedes addition |
| Addition | 35 | Only one operation remains |
Multiplication and division share a priority level, as do addition and subtraction. Within either pair, work from left to right.
A hierarchy with equal-priority pairs
PEMDAS is a memory aid, not six independent priority levels. Multiplication and division have equal priority and are handled from left to right. Addition and subtraction also share a level.
That distinction matters in 24 / 6 x 2. Division and multiplication are evaluated from left to right, giving 4 x 2 = 8.
- Grouping symbols
- Exponents and roots
- Multiplication and division from left to right
- Addition and subtraction from left to right
Ambiguous habits to replace
- Doing multiplication before division regardless of which appears first.
- Treating a fraction bar as though it grouped only the nearest number.
- Assuming -x^2 and (-x)^2 mean the same expression.
Grouping is more than parentheses
Fraction bars, radical bars, absolute-value bars, and function arguments group expressions. In (a+b)/(c+d), both the numerator and denominator are complete grouped expressions.
Nested groups are evaluated from the inside outward. Clear parentheses are preferable when plain-text notation could be read more than one way.
Unary minus and powers
Under standard convention, exponentiation binds before a leading negative sign. Therefore -3^2 means -(3^2) = -9, while (-3)^2 = 9.
Calculator interfaces may display grouping visually, but typed expressions should still include parentheses when the negative number is the base.
-a^2 = -(a^2), while (-a)^2 includes the sign in the baseQuestions about order of operations
Is PEMDAS different from BODMAS?
They describe the same hierarchy; the words differ by region, and multiplication/division still share priority.