Laws of Exponents

Exponents represent repeated multiplication of the same base. For any real number

a

and integer

n > 0

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}.

The properties of exponents allow us to simplify and compare exponential expressions efficiently. These laws apply when the bases are the same or when the exponents follow specific relationships.

Each law stems from basic multiplication and division principles. Understanding why they work helps prevent common errors—especially when simplifying or solving equations that include exponential terms.

Core Skills

Apply the product rule: $a^m \cdot a^n = a^{m+n}$
Apply the quotient rule: $\dfrac{a^m}{a^n} = a^{m-n}$
Apply the power rule: $(a^m)^n = a^{mn}$
Apply the power of a product rule: $(ab)^n = a^n b^n$
Apply the power of a quotient rule: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Key Takeaways

Combine exponents only when the bases match.
Subtract exponents when dividing like bases.
Multiply exponents when taking a power of a power.
Distribute exponents across products or quotients, not sums.
Always check that no bases equal zero when dividing.