Lesson 2: Product of Powers Property

Property

If $a$ is a real number and $m, n$ are counting numbers, then

To multiply with like bases, add the exponents.

Examples

• To simplify $p^3 \cdot p^5$, the bases are the same, so we add the exponents: $p^{3+5} = p^8$.
• In $7^2 \cdot 7^4$, we keep the base $7$ and add the exponents to get $7^{2+4} = 7^6$.
• For $k^8 \cdot k$, remember that $k$ is the same as $k^1$. So, the expression becomes $k^{8+1} = k^9$.

Explanation

When you multiply terms that have the same base, you are just combining their factors.
A simple shortcut is to keep the base the same and just add the exponents together to find the new total.

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.