Order of Operations (BEDMAS)
When a mathematical expression has multiple operations, we need a specific sequence to follow so that everyone arrives at the same answer. This sequence is called the Order of Operations.
The BEDMAS Rule
The acronym BEDMAS helps us remember the correct order:
B
Brackets (or Parentheses)
Calculate everything inside grouping symbols first. Work from the innermost pair outwards. Examples: `( )`, `[ ]`, `{ }`.
Example: $10 - (\mathbf{2+3}) \rightarrow 10 - 5 = 5$
E
Exponents
Evaluate any powers (numbers raised to an exponent).
Example: $5 + \mathbf{3^2} \rightarrow 5 + 9 = 14$
DM
Division & Multiplication
These have equal priority. Perform them as they appear, working from LEFT to RIGHT.
Ex 1 (Division first): $12 \div \mathbf{6 \times 2} \rightarrow \mathbf{2} \times 2 = 4$
Ex 2 (Multiplication first): $10 \times \mathbf{4 \div 5} \rightarrow \mathbf{40} \div 5 = 8$
AS
Addition & Subtraction
These also have equal priority. Perform them as they appear, working from LEFT to RIGHT.
Ex 1 (Subtraction first): $9 - \mathbf{4 + 2} \rightarrow \mathbf{5} + 2 = 7$
Ex 2 (Addition first): $5 + \mathbf{6 - 3} \rightarrow \mathbf{11} - 3 = 8$
Key Points & Common Mistakes:
- Left-to-Right Rule: Always work from left to right when dealing with operations of equal priority (DM or AS).
- Exponents on Negatives: Be careful! $(-3)^2 = (-3) \times (-3) = 9$. The brackets mean the exponent applies to the negative base. BUT $-3^2 = -(3 \times 3) = -9$. Without brackets, the exponent applies only to the 3.
- Implied Multiplication: $2(3+4)$ means $2 \times (3+4)$. Do the brackets first: $2(7) = 14$.
Examples with Step-by-Step Solutions
(The example divs remain the same as in the previous version, using \boxed{} for steps)
Example 1: Simplify $3 + 5 \times 2$
Following BEDMAS:
$$ \begin{align*} & 3 + \boxed{5 \times 2} \quad &\text{(M: Multiplication before Addition)} \\ = & 3 + 10 \\ = & \boxed{13} \quad &\text{(A: Addition)} \end{align*} $$Result: $13$
Example 2: Simplify $(3 + 5) \times 2$
Following BEDMAS:
$$ \begin{align*} & \boxed{(3 + 5)} \times 2 \quad &\text{(B: Brackets first)} \\ = & 8 \times 2 \\ = & \boxed{16} \quad &\text{(M: Multiplication)} \end{align*} $$Result: $16$
Example 3: Simplify $10 - 6 \div 2 + 4^2$
Step-by-step using BEDMAS:
$$ \begin{align*} & 10 - 6 \div 2 + \boxed{4^2} \quad & \text{(E: Exponents)} \\ = & 10 - 6 \div 2 + 16 \\ & 10 - \boxed{6 \div 2} + 16 \quad & \text{(D/M: Division, left-to-right)} \\ = & 10 - 3 + 16 \\ & \boxed{10 - 3} + 16 \quad & \text{(A/S: Subtraction, left-to-right)}\\ = & 7 + 16 \\ & \boxed{7 + 16} \quad & \text{(A/S: Addition, left-to-right)}\\ = & 23 \end{align*} $$Result: $23$
Example 4 (Nested Brackets): Simplify $20 - [12 \div (2 + 4)]$
Work from the innermost brackets outwards:
$$ \begin{align*} & 20 - [12 \div \boxed{(2 + 4)}] \quad & \text{(B: Innermost Bracket)} \\ = & 20 - [12 \div 6] \\ = & 20 - \boxed{[12 \div 6]} \quad & \text{(B: Remaining Bracket - Perform Division inside)} \\ = & 20 - 2 \\ = & \boxed{18} \quad & \text{(S: Subtraction)} \end{align*} $$Result: $18$
Example 5 (Exponents & Negatives): Simplify $5 + (-2)^3 - 3^2$
Watch the exponents carefully:
$$ \begin{align*} & 5 + \boxed{(-2)^3} - \boxed{3^2} \quad & \text{(E: Exponents)} \\ & \quad \text{Note: } (-2)^3 = (-2)(-2)(-2) = -8 \\ & \quad \text{Note: } 3^2 = 3 \times 3 = 9 \\ = & 5 + (-8) - 9 \\ = & 5 - 8 - 9 \\ = & \boxed{5 - 8} - 9 \quad & \text{(A/S: Subtraction, left-to-right)} \\ = & -3 - 9 \\ = & \boxed{-3 - 9} \quad & \text{(A/S: Subtraction)} \\ = & -12 \end{align*} $$Result: $-12$
Example 6 (Fractions & Mixed Operations): Simplify $\left(-\frac{3}{4}\right) \div \frac{1}{5} + \left(\left(-\frac{1}{3}\right) \times \left(-\frac{5}{2}\right)\right)$
Follow BEDMAS with fraction rules:
$$ \begin{align*} & \left(-\frac{3}{4}\right) \div \frac{1}{5} + \boxed{\left(\left(-\frac{1}{3}\right) \times \left(-\frac{5}{2}\right)\right)} \quad & \text{(B: Innermost Bracket - Multiplication)} \\ & \quad \text{Calculation: } (-\frac{1}{3}) \times (-\frac{5}{2}) = +\frac{1 \times 5}{3 \times 2} = \frac{5}{6} \\ = & \left(-\frac{3}{4}\right) \div \frac{1}{5} + \frac{5}{6} \\ = & \boxed{\left(-\frac{3}{4}\right) \div \frac{1}{5}} + \frac{5}{6} \quad & \text{(D/M: Division - Keep, Change, Flip)} \\ & \quad \text{Calculation: } (-\frac{3}{4}) \times \frac{5}{1} = -\frac{3 \times 5}{4 \times 1} = -\frac{15}{4} \\ = & -\frac{15}{4} + \frac{5}{6} \\ = & \boxed{-\frac{15}{4} + \frac{5}{6}} \quad & \text{(A/S: Addition - Find LCD = 12)} \\ & \quad \text{Calculation: } -\frac{15 \times 3}{4 \times 3} + \frac{5 \times 2}{6 \times 2} = -\frac{45}{12} + \frac{10}{12} \\ & \quad = \frac{-45 + 10}{12} = -\frac{35}{12} \\ = & -\frac{35}{12} \end{align*} $$Result: $-\frac{35}{12}$ or $-2\frac{11}{12}$