Lesson 2: Product of Powers Property

Property

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

Examples

• To simplify $z^4 \cdot z^5$, we keep the base $z$ and add the exponents: $z^{4+5} = z^9$.

Property

To raise a power to a power, keep the same base and multiply the exponents. In symbols,

Examples

• To simplify $(x^3)^5$, you multiply the exponents: $x^{3 \cdot 5} = x^{15}$.
• To simplify $(4^2)^3$, you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$.
• Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$, but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$.

Explanation

Think of this as repeated multiplication. $(x^4)^3$ is just $x^4$ multiplied by itself three times. Adding the exponents $4+4+4$ is the same as multiplying $4 \cdot 3$. So, you multiply the exponents.