Lesson 4.5: Order of Operations

New Concept

To avoid confusion in math, we use a specific sequence called the order of operations. This ensures that expressions with multiple steps, like $5 + 2 \cdot 4$, always have one correct answer, no matter who solves them.

What’s next

Now, let's explore these rules one by one. You'll tackle interactive examples and practice problems to master simplifying any expression.

Property

First, perform all multiplications and divisions in order from left to right.

Next, perform all additions and subtractions in order from left to right.

Examples

• To simplify $20 + 4(10) - 15$, first multiply $4(10)$ to get $40$. The expression becomes $20 + 40 - 15$. Then, add and subtract from left to right: $60 - 15 = 45$.

Property

Perform any operations inside parentheses first. Parentheses are used as a grouping symbol to override the standard order of operations when needed.

Examples

• To simplify $25 - 5(8 - 3)$, first perform the subtraction inside the parentheses: $25 - 5(5)$. Then multiply: $25 - 25 = 0$.

• In $36 \div (10 - 4) \cdot 2$, simplify inside the parentheses first: $36 \div 6 \cdot 2$. Then divide and multiply from left to right: $6 \cdot 2 = 12$.

Property

A fraction bar is another kind of grouping symbol. Simplify any expressions above or below a fraction bar before dividing the bottom into the top.

Examples

• To simplify $\frac{20 + 10}{9 - 6}$, first simplify the numerator to $30$ and the denominator to $3$. The expression becomes $\frac{30}{3}$, which equals $10$.

• In $20 - \frac{30}{3(5)}$, first simplify the denominator: $20 - \frac{30}{15}$. Now perform the division: $20 - 2$. Finally, subtract to get $18$.

Property

1. First, perform all operations inside parentheses, or above or below a fraction bar, or inside a radical.
2. Next, compute all powers and roots.
3. Perform all multiplications and divisions in order from left to right.
4. Finally, perform all additions and subtractions in order from left to right.

Examples

• To simplify $10 + 2\sqrt{25}$, first evaluate the root: $10 + 2(5)$. Then multiply: $10 + 10$. Finally, add to get $20$.

• In $\frac{10 + \sqrt{100}}{8 - \sqrt{9}}$, first evaluate the roots: $\frac{10+10}{8-3}$. Then simplify the numerator and denominator: $\frac{20}{5}$. The final answer is $4$.