Lesson 2: Use Multiplication Properties of Exponents

New Concept

This lesson unlocks powerful shortcuts for exponents. You'll learn how to use the Product, Power, and Product to a Power properties to efficiently simplify expressions like $x^2 \cdot x^3$ and $(2y)^4$.

What’s next

Ready to see these rules in action? Next up, you'll walk through several guided examples and then apply your new skills on practice cards.

Property

$a^n$ means multiply $n$ factors of $a$.

($n$ factors)
This is read $a$ to the $n^{\text{th}}$ power. In the expression $a^n$, the exponent tells us how many times we use the base $a$ as a factor.

Examples

• $4^3$ means $4 \cdot 4 \cdot 4$, which simplifies to $64$.
• $(-2)^4$ means $(-2)(-2)(-2)(-2)$, which is $16$. Be careful, as $-2^4$ means $-(2 \cdot 2 \cdot 2 \cdot 2)$, which is $-16$.
• $(\frac{1}{5})^2$ means $\frac{1}{5} \cdot \frac{1}{5}$, resulting in $\frac{1}{25}$.

Explanation

The exponent is a counter for how many times the base number is multiplied by itself.
The base is the number being multiplied, and the exponent is the small number written up high, telling you the count.

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.

Property

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

To raise a product to a power, raise each factor to that power.

Examples

• To simplify $(4x)^2$, the exponent $2$ applies to both the $4$ and the $x$, giving $4^2 x^2 = 16x^2$.
• For $(-3yz)^3$, the exponent applies to each factor: $(-3)^3 y^3 z^3$, which simplifies to $-27y^3z^3$.
• We can combine this with other properties: $(2a^4)^3$ becomes $2^3(a^4)^3 = 8a^{12}$ by using both Product to a Power and Power properties.

Explanation

When multiple items are multiplied inside parentheses and raised to a power, that exponent gets applied to every single item inside.
Think of it as distributing the power to each factor in the product.