Lesson 4: Using Order of Operations

New Concept

Welcome to Algebra! Our journey starts with its most foundational principle: the order of operations, which is a set of rules for simplifying expressions.

What’s next

This lesson builds your foundation. Soon, you'll see these rules in worked examples involving parentheses, exponents, and even comparing expressions.

Property

The rules for simplifying: 1. Grouping Symbols. 2. Powers. 3. Multiply/Divide (left to right). 4. Add/Subtract (left to right).

Examples

$4^3 + 10 \div 2 - 3 \cdot 5 = 64 + 5 - 15 = 54$
$(12 - 5) \cdot 2 + 1 = 7 \cdot 2 + 1 = 14 + 1 = 15$
$45 - (2+4) \cdot 5 = 45 - 6 \cdot 5 = 45 - 30 = 15$

Explanation

Remember PEMDAS! It's the official roadmap for solving expressions. First, handle all operations inside parentheses. Then, simplify any exponents. Next, perform all multiplication and division from left to right. Finally, complete the addition and subtraction, also from left to right. This process ensures one consistent, correct answer.

Property

Exponents are simplified after grouping symbols but before multiplication, division, addition, and subtraction, following the PEMDAS order.

Examples

$5 \cdot 2^3 - 10 = 5 \cdot 8 - 10 = 40 - 10 = 30$
$50 - (1+2)^2 \cdot 3 = 50 - 3^2 \cdot 3 = 50 - 9 \cdot 3 = 50 - 27 = 23$
$4^3 + 9 \div 3 = 64 + 9 \div 3 = 64 + 3 = 67$

Explanation

Exponents get priority after parentheses. This is the 'E' in PEMDAS. Always calculate the value of any powers before you move on to multiplying or dividing. This step simplifies the expression significantly, making the remaining calculations more straightforward and helping to prevent common calculation mistakes down the line.

Property

A fraction bar is a grouping symbol. Simplify the numerator and the denominator completely before the final division.

Examples

$\frac{15 - 3^2 + 4 \cdot 2}{7} = \frac{15 - 9 + 8}{7} = \frac{6 + 8}{7} = \frac{14}{7} = 2$
$\frac{(10 + 2) \cdot 3}{2^2 + 5} = \frac{12 \cdot 3}{4 + 5} = \frac{36}{9} = 4$
$\frac{5^2 - (3+2)}{10} = \frac{25 - 5}{10} = \frac{20}{10} = 2$

Explanation

The fraction bar splits a problem in two. First, use PEMDAS to completely solve the entire expression on top. Then, do the same for the bottom. When you have one number in the numerator and one in the denominator, you can finally divide. It’s like finishing two mini-games before the boss level!