5-2 Numerical Expressions

Property

The term expression means a “phrase that makes sense” made up of numbers, letters, and operations.
A numeric expression is made up of numbers and arithmetic operations.
Two numeric expressions are equivalent if they compute the same number.

Examples

• Evaluate $10 + 4 \cdot 2$. Multiplication comes first: $4 \cdot 2 = 8$. Then add: $10 + 8 = 18$.
• Evaluate $(10 + 4) \cdot 2$. Parentheses come first: $10 + 4 = 14$. Then multiply: $14 \cdot 2 = 28$.
• Show that $3 \cdot 6 + 3 \cdot 2$ is equivalent to $3 \cdot (6 + 2)$. The first expression is $18 + 6 = 24$. The second is $3 \cdot 8 = 24$. Since they both equal 24, they are equivalent.

Explanation

Think of a numeric expression as a recipe. The numbers are your ingredients and the operations $(+, -, \cdot, \div)$ are your cooking steps. Parentheses tell you which steps to do first to get the right delicious result!

Property

When simplifying mathematical expressions perform the operations in the following order:

1. Parentheses and other Grouping Symbols: Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents: Simplify all expressions with exponents.
3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

A common way to remember this is the phrase 'Please Excuse My Dear Aunt Sally'.

Examples

• To simplify $10 - 2 \cdot 3$, you perform multiplication before subtraction: $10 - 6 = 4$.
• To simplify $(10 - 2) \cdot 3$, you perform the operation in parentheses first: $8 \cdot 3 = 24$.
• To simplify $5 + (4-1)^2 \div 3$, you start with parentheses ($5+3^2 \div 3$), then exponents ($5+9 \div 3$), then division ($5+3$), and finally addition, which gives $8$.

Explanation

This is the official rulebook for solving math problems. Following this order (PEMDAS) ensures everyone gets the same correct answer. Always handle groups first, then powers, then multiplication/division, and finally addition/subtraction from left to right.