Exponents and Order of Operations

Exponents

An exponent tells you how many times to multiply the base by itself:

$x^n = \underbrace{x \times x \times \cdots \times x}_{n \text{ times}}$

Examples: $2^3 = 2 \times 2 \times 2 = 8$ $5^2 = 5 \times 5 = 25$ $10^4 = 10{,}000$

Special Exponents

$x^0 = 1 \quad (x \neq 0)$ $x^1 = x$

Order of Operations (PEMDAS)

P - Parentheses (and brackets) E - Exponents M/D - Multiplication and Division (left to right) A/S - Addition and Subtraction (left to right)

Example

$3 + 4 \times 2^2 - (6 + 1)$

1. Parentheses: $3 + 4 \times 2^2 - 7$
2. Exponents: $3 + 4 \times 4 - 7$
3. Multiplication: $3 + 16 - 7$
4. Left to right: $19 - 7 = 12$

Common Mistakes

❌ $2 + 3 \times 4 = 20$ (wrong — did addition first) ✅ $2 + 3 \times 4 = 2 + 12 = 14$ (correct — multiplication first)

❌ $-3^2 = 9$ (wrong) ✅ $-3^2 = -(3^2) = -9$ (the exponent only applies to the 3) ✅ $(-3)^2 = 9$ (now the exponent applies to $-3$)

Memory aid: "Please Excuse My Dear Aunt Sally" for PEMDAS.