# Products of Exponents (Product Rule) Worksheets

We explore how to approach multiplying pairs of exponential values that share the same base number. When you come across an example like this: m^{4} x m^{2}. The product rule of exponents tells us that if we are multiplying exponents with the same base, we simply add the exponents to determine the product. This means that m^{4} x m^{2} = m^{6 (4 + 2)}. This is a nice rule that is helpful to speed up our calculations especially when condensing expressions and equations. These worksheets and lessons help students learn how to multiply values that have attached exponents.

### Aligned Standard: 8.EE.A.1

- Step-by-Step Lesson- There are two levels of difficulty presented with the lesson.
- Guided Lesson - We advance towards products with binomials.
- Guided Lesson Explanation - Each problem is deeply thought through for you.
- Practice Worksheet - The here is with simple products and negatives are not really the focus.
- Matching Worksheet - Make those problems pop and match the expressions to their end product.

- Answer Keys - These are for all the unlocked materials above.

### Multiplying Positive Exponents Sheets

We start easy with monomials.

- Sheet 1 - Solve the following problems.
- Sheet 2 - Find the products of values that contain exponents.
- Sheet 3 - These contain binomials.

### Multiple Variable Product Exponents Worksheets

The difficulty progresses here. Remember your answers should only include positive exponents.

- Practice 1 - These will take some time hang it out there.
- Practice 2 - You will find fractional values.
- Practice 3 - Trinomials start to show up here.
- Practice 4 - You will find a mixed values appear here.

### Product Rule with Variables Worksheets

Simplify using the product rule for exponents.

- Practice 1 - Negative exponent will appear here.
- Practice 2 - We introduce parentheses in these at this point.
- Practice 3 - Hint: All of the answers will have positive exponents.
- Practice 4 - A mix of negative exponents do a wild mix of problems.

### Multiple Digit Product Rule Problems Sheets

This is your basic simple products with additional an additional exponential value.

- Sheet 1 - Tons of fractions are jumped around.
- Sheet 2 - Apply the product rule to solve the problems.
- Sheet 3 - Just add up those exponents.

### Find the Product of Positive and Negative Exponents Sheets

You will also need to pay attention to the non-exponential value as well.

- Sheet 1 - Simple binomials are found up here.
- Sheet 2 - Some unknown variables and some not.
- Sheet 3 - All problems have the same variable inside.

### Basic Product Rule Form

The work off of the practice sheet that was first presented above.

- Practice 2 - No variables at all.
- Practice 3 - Just bigger values.

### What is the Product Rule of Exponents?

Thanks to the shorthand rule in mathematics, we don’t have to write long numbers that can literally take up the whole page. Exponents are a shorthand method followed in mathematics to write up repeated multiplication of the same number by itself. The variable or number that is being multiplied is called the base number, and the number of times it’s being multiplied is written as many times as the exponent is. The exponent concept is not new to many of us. However, the product rule of exponents might be something new for many of us. Consider the following example:

The product of x^{3} . x^{4} = x . x . x (3 factors) multiplied by x . x . x . x (4 factors).

In the above example, you can find a cube of x and 4 times of x being multiplied. As per the product rule of exponent, when the bases of two exponential numbers are the same, you can add the exponents of the two numbers. Like in this case, you can add the powers/exponents of x which are 3 and 4. Hence, the answer becomes x^{3 + 7} = x^{ 10}. This is the product rule of an exponent.

The product rule can be visualized the formula: a^{b} x a^{c} = a^{b + c}

This holds true not just for individual values, but whole exponential expressions as well.