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.

Solved Examples

Jump directly to a worked example by expression type.

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 $3+2(5^2-9)$.

  1. Test the expression and identify the innermost grouping.
  2. Evaluate the power inside the parentheses.
  3. Evaluate the subtraction inside the parentheses.
  4. Multiply by the outside coefficient.
  5. Add and state the simplified expression: $3+2(5^2-9)=35$.
$$3+2(5^2-9)$$ $$=3+2(25-9)$$ $$=3+2(16)$$ $$=3+32$$ $$=35$$

Left to Right

Simplify $24\div 3\cdot 2$.

  1. Multiplication and division have equal priority, so begin at the left.
  2. Divide: $24\div3=8$.
  3. Multiply and state the simplified expression: $24\div3\cdot2=16$.
$$24\div 3\cdot 2$$ $$=8\cdot 2$$ $$=16$$

Fraction Bar Grouping

Simplify $\dfrac{8+4^2}{6-2}$.

  1. Treat the numerator and denominator as grouped expressions.
  2. Evaluate the exponent in the numerator and simplify the denominator.
  3. Simplify the numerator: $8+16=24$.
  4. Divide and state the simplified expression: $\frac{8+4^2}{6-2}=6$.
$$\frac{8+4^2}{6-2}$$ $$=\frac{8+16}{4}$$ $$=\frac{24}{4}$$ $$=6$$

Negative Sign with Powers

Compare $(-4)^2$ and $-4^2$.

  1. Parentheses make $-4$ the base, so $(-4)^2=16$.
  2. Without parentheses, the exponent applies to $4$ first, so $-4^2=-16$.
$$(-4)^2=(-4)(-4)=16$$ $$-4^2=-(4^2)=-16$$

Nested Grouping

Simplify $2[18-(3+4)^2]+5$.

  1. Start with the innermost parentheses: $3+4=7$.
  2. Evaluate the power: $7^2=49$.
  3. Subtract inside the brackets: $18-49=-31$.
  4. Multiply by the outside coefficient: $2(-31)=-62$.
  5. Add $5$.
  6. State the simplified expression: $2[18-(3+4)^2]+5=-57$.
$$2[18-(3+4)^2]+5$$ $$=2[18-7^2]+5$$ $$=2[18-49]+5$$ $$=2[-31]+5$$ $$=-62+5$$ $$=-57$$

Absolute Value

Simplify $|-6+2^3|-5$.

  1. Treat absolute value as grouping and evaluate the power inside it.
  2. Simplify inside the absolute-value bars: $-6+8=2$.
  3. Evaluate the absolute value: $|2|=2$.
  4. Subtract $5$.
  5. State the simplified expression: $|-6+2^3|-5=-3$.
$$|-6+2^3|-5$$ $$=|-6+8|-5$$ $$=|2|-5$$ $$=2-5$$ $$=-3$$

Distribution Comes from Multiplication

Simplify $4-3(2-5)^2$.

  1. Evaluate the grouping: $2-5=-3$.
  2. Evaluate the power: $(-3)^2=9$.
  3. Multiply by the outside coefficient: $3(9)=27$.
  4. Subtract from $4$.
  5. State the simplified expression: $4-3(2-5)^2=-23$.
$$4-3(2-5)^2$$ $$=4-3(-3)^2$$ $$=4-3(9)$$ $$=4-27$$ $$=-23$$