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Product Rule
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The word product means to multiply. The product rule of exponents helps us remember what we do when two numbers with exponents are multiplied together. Here are some math vocabulary words that will help you to understand this lesson better:

• Base = the number or variable that is being multiplied to itself.
• Power = the number of the exponent, how many times the base is multiplied to itself.

The following video will explain, with some examples, what the product rule is; the word product in math means what you get when you multiply things together.
$$\text{x}^3\,\text{x}^4 = \text{x}^7$$

Rules of Exponents-Product Rule

Qualification for the product rule: bases must be the same. If the bases are the same, then the product rule says that you add the exponents. Also remember that you can multiply in any order, so $$(a)(b) = (b)(a)$$. This means that if there are multiple bases, you can rearrange the order and add the exponents of any of the bases that are the same.

### Practice Problems

Simplify the following Expressions:
1. $$\text{x}\,{\text{x}}^{3}$$ (
Solution
Solution:
$${\text{x}}^{4}$$
Details:
The product rule states that if two factors raised to an exponent are being multiplied together, and they have the same base, we can add the exponents.

In this example, $${\text{x}}$$ and $${\text{x}}^{3}$$ are our two factors. Factors are the numbers that multiply together to make another number or expression.

A number without an exponent is the same as a number with exponent 1.

$${\text{x}} = {\text{x}}^{1}$$

$${\color{Blue} {\text{x}}}^{{\color{Red} ?}}\,{\color{DarkGreen} {\text{x}}^{3}}\rightarrow {\color{Blue} {\text{x}}^{1}}\,{\color{DarkGreen} {\text{x}}^{3}}$$

We can rewrite our example problem as $${\text{x}}^{1}\,{\text{x}}^{3}$$

So $${\text{xx}}^{3} = {\text{x}}^{1}\,{\text{x}}^{3}$$

In this example, both factors have the same base x, so we can add the exponents together.

$${\text{x}}^{1}\,{\text{x}}^{3} = {\text{x}}^{\left ( 1 + 3 \right )} = {\text{x}}^{4}$$

Another way to look at this is to examine what the factors mean.

$${\text{x}}^{3}$$ Is the same as x multiplied together 3 times.

$${\color{Blue} {\text{x}}}\,{\color{DarkGreen} {\text{x}}^{3}}= {\color{Blue} {\text{x}}}\,{\color{DarkGreen} {\text{xxx}}}$$

So $${\text{xx}}^{3} = {\text{xxxx}}$$

This is a total of 4 x’s multiplied together. We can rewrite that as $${\text{x}}^{4}$$.
)
2. $${\text{m}}^{2}\,{\text{m}}^{5}$$ (
Solution
Solution:
$${\text{m}}^{7}$$
)
3. $${\text{m}}^{3}\,{\text{x}}^{6}\,{\text{m}}$$ (
Video Solution
Solution:
$${\text{m}}^{4}\,{\text{x}}^{6}$$
Details:

(Video Source | Transcript)
)
4. $${\text{x}}^{2}\,{\text{y}}^{3}\,{\text{x}}^{4}$$ (
Solution
Solution:
$${\text{x}}^{6}\,{\text{y}}^{3}$$
)
5. $${\text{mx}}^{2}\,{\text{m}}^{3}\,{\text{x}}^{7}$$ (
Solution
Solution:
$${\text{m}}^{4}\,{\text{x}}^{9}$$
Details:
Below there are two different ways to solve the same problem.

Version 1:

In this example, there are four factors: $${\text{m}}$$, $${\text{x}}^{2}$$, $${\text{m}}^{3}$$, and $${\text{x}}^{7}$$.

We don’t see an exponent on the factor m, but this actually means it has an exponent of 1.

Since everything is being multiplied together, we can rearrange the factors so the m factors are next to each other and the x factors are next to each other.

$${\text{m}}^{1}\,{\text{x}}^{2}\,{\text{m}}^{3}\,{\text{x}}^{7} = {\text{m}}^{1}\,{\text{m}}^{3}\,{\text{x}}^{2}\,{\text{x}}^{7}$$

Our final answer is $${\text{m}}^{4}\,{\text{x}}^{9}$$

Version 2:

Another way to look at this is to break all the factors with exponents into the multiplication of their bases.

m multiplied 1 time

x multiplied 2 times

m multiplied 3 times

x multiplied 7 times

$${\text{mxxmmmxxxxxxx}}$$

Since all the factors are being multiplied together we can rearrange them so the factors with the same base are next to each other.

$${\text{mxxmmmxxxxxxx = mmmmxxxxxxxxx}}$$

We see there are 4 m factors being multiplied together and 9 x factors being multiplied together. The final step is to rewrite this in exponent form.

Our final answer is $${\text{m}}^{4}\,{\text{x}}^{9}$$.
)
6. $${\text{x}}^{5}\,{\text{y}}^{4}\,{\text{x}}^{2}\,{\text{y}}\,{\text{z}}^{2}$$ (
Video Solution
Solution:
$${\text{x}}^{7}\,{\text{y}}^{5}\,{\text{z}}^{2}$$
Details:

(Video Source | Transcript)
)

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