Exponent Product Rule — Definition & Examples

(Video) Product Rule for Exponents

by mathman1024

Here's a video by mathman1024 explaining the product rule.

$\large{\text{Summary}}$

The product rule for exponents states: $$x^a\cdot x^b=x^{a+b}$$

We're going to focus on the rule when $a$ and $b$ are natural numbers.

To see why, let's look at the example $x^2\cdot x^3$. If we expand these we see

$$x^2\cdot x^3=(x\cdot x)\cdot(x\cdot x\cdot x)$$

The first term has two factors of $x$ and the second term has three factors of $x$. When we multiply these two we see there are now five factors of $x$.

$$\begin{align}x^2\cdot x^3&=(x\cdot x)\cdot(x\cdot x\cdot x)\\ &=x\cdot x\cdot x\cdot x\cdot x\\ &=x^5 \end{align}$$

So this rule tells us that if we multiply two exponents with the same base then we just add the exponents together.

Let's try another example: $x^5\cdot x^8$. $$\begin{align} x^5\cdot x^8&=x^{5+8}\\ &=x^{13} \end{align}$$

Try these other examples on your own and then watch the video to see if you get the same answers. Notice that this rule applies for variables and for numbers.

1. $x^{15}\cdot x^{10}$
2. $x^{43}\cdot x^{54}$
3. $3^7\cdot3^4$
4. $x^{14}y^5\cdot x^{11}y^9$

Adding Exponents

Let's be honest—exponents are little tiny numbers that can cause enormous headaches if you don't know how to handle them.

The good news is that there are a few simple rules we can follow that help turn exponents into nothing more than cute mini-numbers sitting on the shoulders of their regular-sized friends.

Let's quickly summarize the parts of an exponential function before we move on. Take a look at this picture:

In this thread, we are going to review the Product of Powers Rule. This rule allows us to add exponents when their bases are the same.

Here's how the Product of Powers Rule can be translated: When the bases of exponential functions are the same and are being multiplied together, leave the base alone and just add the exponents together.

Here's an example using actual numbers:

Let's try some examples.

(Video) Basic Exponent Properties

by PatrickJMT

This video by PatrickJMT explains the basic properties of exponents. Recall that an exponent is just a shorthand way of referring to repeated multiplication. $$5^3=5\cdot5\cdot5=125$$

$\large{\text{Summary}}$

Product Rule

The product rule says that if we have exponents with the same base being multiplied, then we can just add the exponents together. $$\boxed{a^m\cdot a^n=a^{m+n}}$$

For example, suppose we wanted to simplify $3^4\cdot3^2$. Our rule tells us: $$3^4\cdot3^2=3^{4+2}=3^6$$ And to prove this we can write out the multiplication. $$\begin{align} 3^4\cdot3^2&=(3\cdot3\cdot3\cdot3)\cdot(3\cdot3)\\ &=3\cdot3\cdot3\cdot3\cdot3\cdot3\\ &=3^6 \end{align}$$

This rule is not to be confused with the power rule. In that exponent rule, we multiply powers together, rather than add them.