## The Power Rule for Exponents

### Learning Outcome

• Simplify exponential expressions with like bases using the product, quotient, and power rules

## Raise Powers to Powers

Another word for exponent is power. You have likely seen or heard an example such as $3^5$ can be described as $3$ raised to the $5$th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power and what to do when two numbers or variables are multiplied and both are raised to a power. We will also learn what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.

Let us simplify $\left(5^{2}\right)^{4}$. In this case, the base is $5^2$ and the exponent is $4$, so you multiply $5^{2}$ four times: $\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}$ (using the Product Rule—add the exponents).

$\left(5^{2}\right)^{4}$ is a power of a power. It is the fourth power of $5$ to the second power. We saw above that the answer is $5^{8}$. Notice that the new exponent is the same as the product of the original exponents: $2\cdot4=8$.

So, $\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}$ (which equals 390,625 if you do the multiplication).

Likewise, $\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}$

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, $\left(2^{3}\right)^{5}=2^{15}$.

### The Power Rule for Exponents

For any positive number x and integers a and b: $\left(x^{a}\right)^{b}=x^{a\cdot{b}}$.

Take a moment to contrast how this is different from the product rule for exponents found on the previous page.

### Example

Write each of the following products with a single base. Do not simplify further.

1. ${\left({x}^{2}\right)}^{7}$
2. ${\left({\left(2t\right)}^{5}\right)}^{3}$
3. ${\left({\left(-3\right)}^{5}\right)}^{11}$

In the following video, you will see more examples of using the power rule to simplify expressions with exponents.

Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power and typically contained within parentheses. In this case, you multiply the exponents.

Product Rule Power Rule
$5^{3}\cdot5^{4}$ =  $5^{3+4}$ = $5^{7}$ but $\left(5^{3}\right)^{4}$ = $5^{3\cdot4}$ = $5^{12}$
$x^{5}\cdot x^{2}$ = $x^{5+2}$ = $x^{7}$ but $\left(x^{5}\right)^{2}$ =  $x^{5\cdot2}$ = $x^{10}$
$\left(3a\right)^{7}\cdot\left(3a\right)^{10}$ = $\left(3a\right)^{7+10}$ = $\left(3a\right)^{17}$ but $\left(\left(3a\right)^{7}\right)^{10}$ = $\left(3a\right)^{7\cdot10}$ = $\left(3a\right)^{70}$