Laws of Exponents — The Core Rules

Exponents (also called powers or indices) follow a set of consistent rules that apply whether the exponents are integers, fractions, or any real numbers. Mastering these three laws is essential for simplifying expressions and solving equations throughout the course.

The three fundamental laws are:

Product Rule: $a^{x} \cdot a^{y} = a^{x + y}$
When multiplying powers with the same base, add the exponents.
Quotient Rule: $\frac{a ^{x}}{a ^{y}} = a^{x - y}$
When dividing powers with the same base, subtract the exponents.
Power Rule: $(a^{x})^{y} = a^{x y}$
When raising a power to another power, multiply the exponents.

Note

All three rules require the same base. You cannot directly apply these rules to expressions like $2^{3} \cdot 3^{3}$ using the product rule — the bases are different.

Example

Product Rule: $4^{6} \times 4^{8} = 4^{6 + 8} = 4^{14}$

Quotient Rule: $\frac{5 ^{4}}{5 ^{2}} = 5^{4 - 2} = 5^{2} = 25$

Power Rule: $(4^{2})^{5} = 4^{2 \times 5} = 4^{10}$

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SL 1.5—Exponents and Logarithms

Laws of Exponents — The Core Rules

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