Exponent Rules

Basic Exponent Notation

$a^n = a \cdot a \cdot a \cdots a \text{ (n times)}$

Example: $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$

Product Rule

When multiplying with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$

Example: $x^3 \cdot x^5 = x^{3+5} = x^8$

Quotient Rule

When dividing with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$

Example: $\frac{x^7}{x^3} = x^{7-3} = x^4$

Power Rule

When raising a power to a power, multiply the exponents: $(a^m)^n = a^{mn}$

Example: $(x^2)^3 = x^{2 \cdot 3} = x^6$

Power of a Product

$(ab)^n = a^n b^n$

Example: $(2x)^3 = 2^3 x^3 = 8x^3$

Power of a Quotient

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Zero Exponent

$a^0 = 1 \text{ (for } a \neq 0 \text{)}$

Negative Exponents

$a^{-n} = \frac{1}{a^n}$

Example: $x^{-3} = \frac{1}{x^3}$