Lesson 52: Order of Operations

New Concept

Welcome! This course builds your math skills step-by-step. We begin with the fundamental rules that ensure mathematical language is clear and consistent for everyone.

What’s next

Our journey starts with the Order of Operations, the universally accepted sequence for solving problems. Soon, you'll tackle worked examples to see these rules in action.

Property

1. Simplify within parentheses (or other symbols of inclusion).
2. Simplify powers and roots.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.

Examples

$2 + 4 \times 3 - 4 \div 2 = 2 + 12 - 2 = 12$
$5 + 5 \cdot 5 - 5 \div 5 = 5 + 25 - 1 = 29$
$50 - 8 \cdot 5 + 6 \div 3 = 50 - 40 + 2 = 12$

Explanation

Think of 'Please Excuse My Dear Aunt Sally' as the ultimate rulebook for math! Following this sequence prevents mathematical chaos and ensures everyone arrives at the same correct answer. It is the essential guide to solving expressions correctly, no matter how complicated they might look at first.

Property

Symbols of inclusion set apart portions of an expression so they may be evaluated first. The most common symbols of inclusion are parentheses ( ), brackets [ ], braces { }, and the division bar in a fraction.

Examples

$4 \times (3 + 7) = 4 \times 10 = 40$
$\frac{9^2 + 3 \cdot 5}{2} = \frac{81 + 15}{2} = \frac{96}{2} = 48$
$\frac{2^3 + 3^2 + 2 \cdot 5}{3} = \frac{8 + 9 + 10}{3} = \frac{27}{3} = 9$

Explanation

These symbols are like a VIP pass in a math problem, telling you to 'Solve this part first!' Always handle the math inside these groups, like what is in parentheses or above and below a fraction bar, before you move on to anything else in the expression.

Property

To evaluate an expression with variables, first substitute the given numerical values for each variable. Then, you must apply the order of operations to simplify the resulting numerical expression and calculate the final value.

Examples

Evaluate $a + ab$ if $a = 3$ and $b = 4$: $(3) + (3)(4) = 3 + 12 = 15$
Evaluate $xy - \frac{x}{2}$ if $x = 9$ and $y = \frac{2}{3}$: $(9)(\frac{2}{3}) - \frac{(9)}{2} = 6 - 4.5 = 1.5$
Evaluate $ab - bc$ if $a = 5, b = 3, c = 4$: $(5)(3) - (3)(4) = 15 - 12 = 3$

Explanation

Ready to bring variables to life? First, swap each letter for its given number. A pro tip is using parentheses when you plug the numbers in to prevent mix-ups. After that, just unleash your order of operations skills on the expression to find the final answer!