Order of Operations with Exponents

When evaluating mathematical expressions that have multiple operations, you must follow a specific set of rules to get the correct answer. This rule is commonly known as PEMDAS.

The PEMDAS Rule

PEMDAS tells you the exact order to solve parts of an expression:

1. Parentheses: Solve anything inside parentheses () first.
2. Exponents: Calculate powers and roots (like $2^2$ or $3^3$).
3. Multiplication and Division: Solve these from left to right.
4. Addition and Subtraction: Solve these from left to right.

The key addition here is Exponents. Always evaluate exponents after parentheses but before you multiply or divide.

Step-by-Step Examples

Let's apply PEMDAS to a few expressions.

Example 1: Evaluate $3 + 2^2 \times 4$

• Step 1 (Exponents): Calculate $2^2$. Since $2^2 = 4$, the expression becomes $3 + 4 \times 4$.
• Step 2 (Multiplication): Multiply $4 \times 4 = 16$. The expression becomes $3 + 16$.
• Step 3 (Addition): Add $3 + 16 = 19$.
• Answer: $19$

Example 2: Calculate $(5 - 2)^3 + 4 \times 2$

• Step 1 (Parentheses): Solve inside the parentheses: $5 - 2 = 3$. The expression becomes $3^3 + 4 \times 2$.
• Step 2 (Exponents): Calculate $3^3$. Since $3 \times 3 \times 3 = 27$, the expression becomes $27 + 4 \times 2$.
• Step 3 (Multiplication): Multiply $4 \times 2 = 8$. The expression becomes $27 + 8$.
• Step 4 (Addition): Add $27 + 8 = 35$.
• Answer: $35$

Example 3: Evaluate $2 \times 3^2 - (4 + 1)$

• Step 1 (Parentheses): Solve $4 + 1 = 5$. The expression becomes $2 \times 3^2 - 5$.
• Step 2 (Exponents): Calculate $3^2 = 9$. The expression becomes $2 \times 9 - 5$.
• Step 3 (Multiplication): Multiply $2 \times 9 = 18$. The expression becomes $18 - 5$.
• Step 4 (Subtraction): Subtract $18 - 5 = 13$.
• Answer: $13$