Section 10.2: Product of Powers Property

Property

Any repeated multiplication can be written in exponential form: $a \cdot a \cdot a \cdot \ldots \cdot a$ ($n$ factors) = $a^n$.

Conversely, any exponential expression can be expanded back into repeated multiplication. The base ($a$) indicates what number is being multiplied, and the exponent ($n$) indicates how many times the base appears as a factor.

Examples

• $5 \cdot 5 \cdot 5 \cdot 5 = 5^4$ (four factors of 5)
• $x^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x$ (six factors of $x$)
• $(-3) \cdot (-3) \cdot (-3) = (-3)^3$ (three factors of -3)

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

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

To raise a power to a power, multiply the exponents.

Examples

• To simplify $(y^6)^3$, we are raising a power to a power, so we multiply the exponents: $y^{6 \cdot 3} = y^{18}$.
• For $(4^3)^5$, we use the power property to get $4^{3 \cdot 5} = 4^{15}$.
• You can see this by expanding: $(x^2)^4$ is $x^2 \cdot x^2 \cdot x^2 \cdot x^2$, which equals $x^{2+2+2+2} = x^8$. The shortcut is $x^{2 \cdot 4} = x^8$.

Explanation

Raising a power to another power means you have a group of factors repeated multiple times.
The quickest way to find the total number of factors is to multiply the inner exponent by the outer exponent.