Foundations Reference
Order of Operations
Order of operations is the grammar of arithmetic. It tells us which part of an expression to simplify first so the same expression has one agreed meaning.
Fact Table
| Priority | Operation | Reminder |
|---|---|---|
| 1 | Grouping symbols. | Work inside parentheses, brackets, fraction bars, radicals, and absolute value first. |
| 2 | Exponents and roots. | Powers apply before multiplication unless grouping says otherwise. |
| 3 | Multiplication and division. | Same priority. Work left to right. |
| 4 | Addition and subtraction. | Same priority. Work left to right. |
| Always | Negative signs. | Track whether the negative sign is part of the base or outside a power. |
Content Formulas
Main Order
$$\text{Group} \rightarrow \text{Powers} \rightarrow \text{Multiply/Divide} \rightarrow \text{Add/Subtract}$$
Left to Right
$$\frac{24}{3}\cdot 2=8\cdot 2=16$$
Negative Base
$$(-3)^2=9$$
Negative Outside
$$-3^2=-(3^2)=-9$$
PEMDAS is useful, but the real rule is that multiplication and division are tied, and addition and subtraction are tied. For each tied pair, move left to right.
Classic Examples
Parentheses First
Simplify $3+2(5^2-9)$.
Operation MoveInside the parentheses, powers still happen before subtraction.
$$3+2(5^2-9)=3+2(25-9)$$
$$3+2(5^2-9)=3+2(16)$$
$$3+2(5^2-9)=3+32$$
$$3+2(5^2-9)=35$$
Left to Right
Simplify $24\div 3\cdot 2$.
Operation MoveDivision and multiplication are tied, so work from left to right.
$$24\div 3\cdot 2=8\cdot 2$$
$$24\div 3\cdot 2=16$$
Fraction Bar Grouping
Simplify $\dfrac{8+4^2}{6-2}$.
Operation MoveA fraction bar groups the entire numerator and the entire denominator.
$$\frac{8+4^2}{6-2}=\frac{8+16}{4}$$
$$\frac{8+4^2}{6-2}=\frac{24}{4}$$
$$\frac{8+4^2}{6-2}=6$$
Negative Sign with Powers
Compare $(-4)^2$ and $-4^2$.
Operation MoveParentheses decide whether the negative sign is part of the base.
$$(-4)^2=(-4)(-4)=16$$
$$-4^2=-(4^2)=-16$$
Nested Grouping
Simplify $2[18-(3+4)^2]+5$.
Operation MoveWork from the innermost grouping outward.
$$2[18-(3+4)^2]+5=2[18-7^2]+5$$
$$2[18-(3+4)^2]+5=2[18-49]+5$$
$$2[18-(3+4)^2]+5=2[-31]+5$$
$$2[18-(3+4)^2]+5=-62+5$$
$$2[18-(3+4)^2]+5=-57$$
Absolute Value
Simplify $|-6+2^3|-5$.
Operation MoveAbsolute value acts like grouping; simplify inside it first, then take distance from zero.
$$|-6+2^3|-5=|-6+8|-5$$
$$|-6+2^3|-5=|2|-5$$
$$|-6+2^3|-5=2-5$$
$$|-6+2^3|-5=-3$$
Distribution Comes from Multiplication
Simplify $4-3(2-5)^2$.
Operation MoveSimplify the grouping, then the power, then multiply by the outside coefficient.
$$4-3(2-5)^2=4-3(-3)^2$$
$$4-3(2-5)^2=4-3(9)$$
$$4-3(2-5)^2=4-27$$
$$4-3(2-5)^2=-23$$