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# Product Rules for Exponents Worksheet, Definition, and Examples

Get the free Product Rule for Exponents worksheet and other resources for teaching & understanding solving the Product Rule for Exponents

Multiplying Exponents Rules (Do you add Exponents when Multiplying Exponents with the base?)

### Key Points about the Product Rule

• The product rule for exponents is used to simplify expressions with exponents by adding their exponents.
• When multiplying exponential expressions with variables, we need to use the product rule for exponents to simplify them.
• The product rule for exponents is a powerful tool in algebra that allows us to simplify complex expressions with ease and solve algebraic equations efficiently.

## How to Multiply Exponents

The exponent rule for multiplying exponential terms together is called the Product Rule. The Product Rule states that when multiplying exponential terms together with the same base, you keep the base the same and then add the exponents. If the exponential terms have multiple bases, then you treat each base like a common term. That means that only the bases that are the same will be multiplied together.

The product rule for exponents is a fundamental concept in algebra that allows us to simplify expressions with exponents. Exponents are used to represent repeated multiplication of a number or variable, and the product rule tells us how to multiply two exponential expressions with the same base. The product rule states that when multiplying two exponential expressions with the same base, we add their exponents. This rule is essential in simplifying expressions with exponents and is used extensively in algebra.

Multiplying exponents with the same base is straightforward and involves adding their exponents. However, when multiplying exponential expressions with variables, we need to use the product rule for exponents to simplify them. The product rule for exponents is used when multiplying exponential expressions with the same base and involves adding their exponents. This rule is essential in simplifying expressions with multiple variables raised to different powers.

The product rule for exponents is a powerful tool in algebra that allows us to simplify expressions with exponents. By understanding this rule, we can simplify complex expressions with ease and solve algebraic equations efficiently. In the next section, we will look at some examples of how the product rule for exponents is used in algebra.

Common Core Standard: 8.EE.A.1
Basic Topics:
Related Topics: Quotient Rule, Power of a Power Rule, Power of a Quotient Rule, Power of a Product Rule, Negative Exponents

## Multiplying Exponents with Same Base

In mathematics, the product rule for exponents is used to multiply two exponential terms with the same base. When two exponential terms have the same base, the base remains the same, and the exponents are added. The product rule for exponents can be stated as follows:

a^m * a^n = a^(m+n)

Here, a represents the base, and m and n represent the exponents. When two exponential terms with the same base are multiplied, the result is a new exponential term with the same base and a new exponent equal to the sum of the exponents in the original terms.

For example, if we multiply 2^3 and 2^4, we get:

2^3 * 2^4 = 2^(3+4) = 2^7

Multiplying exponential terms with the same base is a fundamental concept in algebra. It is used in many different areas of mathematics, including calculus, geometry, and trigonometry.

The product rule for exponents can also be extended to include more than two exponential terms with the same base. When multiplying three or more exponential terms with the same base, we simply add all the exponents together. For example:

2^2 * 2^3 * 2^4 = 2^(2+3+4) = 2^9

In conclusion, when multiplying exponential terms with the same base, we can use the product rule for exponents to simplify the expression. The base remains the same, and the new exponent is equal to the sum of the exponents in the original terms. This concept is fundamental in algebra and is used in many different areas of mathematics.

## Multiplying Exponents with Variables

When multiplying exponents with variables, the product rule can be used to simplify the expression. The rule states that when multiplying two exponential expressions with the same base, the exponents can be added together while keeping the base the same.

For example, consider the expression `x^2 * x^3`. Since both terms have the same base `x`, the exponents can be added together to get `x^(2+3) = x^5`. Therefore, the product of `x^2` and `x^3` is `x^5`.

It is important to note that when multiplying exponential expressions with different bases, the product rule cannot be used. In such cases, the expressions must be simplified using other rules of exponents.

When multiplying exponential expressions with variables and coefficients, the same product rule can be used. The coefficients can be multiplied together, and the variables with the same base can be added together while keeping the base the same.

For example, consider the expression `3x^2 * 2x^3`. The coefficients `3` and `2` can be multiplied together to get `6`. The variables `x^2` and `x^3` have the same base `x`, so the exponents can be added together to get `x^(2+3) = x^5`. Therefore, the product of `3x^2` and `2x^3` is `6x^5`.

In summary, when multiplying exponential expressions with variables, the product rule can be used to simplify the expression by adding the exponents of the same base together while keeping the base the same. When coefficients are involved, they can be multiplied together, and the variables with the same base can be added together while keeping the base the same.

## Multiplying Exponents with Different Bases

When multiplying exponents with different bases, the product rule for exponents applies. The rule states that when multiplying two exponential expressions with different bases, the bases remain the same, and the exponents are added together.

For example, when multiplying 2^3 and 3^2, the product rule states that the result is (2*3)^(3+2), which simplifies to 6^5.

To understand this rule better, it can be helpful to expand the expressions into their base form before multiplying. For instance, 2^3 can be expanded to 222, and 3^2 can be expanded to 33. Multiplying these two expressions results in (222)(33), which can be further simplified to 2223*3. This expression can then be written in exponential form as 6^2, which is the same as 6^5 using the product rule.

It is important to note that this rule only applies when the bases are different. When the bases are the same, the exponents can be added together directly.

In summary, when multiplying exponents with different bases, the bases remain the same, and the exponents are added together using the product rule for exponents. It can be helpful to expand the expressions into their base form before multiplying to better understand the rule.

## 3 Simple Product Rule for Exponents Examples

The product rule for exponents is a fundamental rule in mathematics that states that when multiplying two expressions with the same base, add the exponents. This rule is essential in simplifying expressions with multiple variables and exponents.

1. Identify the terms that have the same base.
2. If the bases are the same, you will add the exponents of the bases together.
3. If the bases are different, you will keep the exponents separate.
4. If an exponents is negative, be sure to include the negative when adding.

Here are a few examples to help illustrate the product rule for exponents:

### Example 1

Simplify the expression: 2^3 * 2^5

Using the product rule for exponents, we can add the exponents to get:

2^3 * 2^5 = 2^(3+5) = 2^8

Therefore, the simplified expression is 256.

### Example 2

Simplify the expression: (3x^2)^4 * (3x^2)^3

Using the product rule for exponents, we can add the exponents to get:

(3x^2)^4 * (3x^2)^3 = 3^(4+3) * x^(24+23) = 3^7 * x^14

Therefore, the simplified expression is 2187x^14.

### Example 3

Simplify the expression: (2^3 * 3^2)^2

Using the product rule for exponents, we can multiply the exponents by the power outside the parentheses to get:

(2^3 * 3^2)^2 = 2^(32) * 3^(22) = 2^6 * 3^4

Therefore, the simplified expression is 2304.

These examples demonstrate how the product rule for exponents can be used to simplify expressions with multiple variables and exponents. To practice using this rule, students can complete exercises and worksheets that provide additional problems to solve.

## 5 Quick Product Rule Practice Problems

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Product Rule Quiz

Click Start to begin the practice quiz!

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Simplify using the Product Rule.

3^-1 x 3^5

2 / 5

Simplify using the Product Rule.

7^-3 x 7^11

3 / 5

Simplify using the Product Rule.

5^40 x 5^-5

4 / 5

Simplify using the Product Rule.

8^4 x 8^11

5 / 5

Simplify using the Product Rule.

9^2 x 9^3

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## Product Rule Definition Algebra

The product rule for exponents is a fundamental rule of algebra that is used to simplify expressions with the same base. For any real number a and positive numbers m and n, the product rule for exponents states that:

a^m * a^n = a^(m+n)

This means that when you multiply two expressions with the same base, you add the exponents. For example, if you have 2^3 * 2^4, you can use the product rule for exponents to simplify it as:

2^3 * 2^4 = 2^(3+4) = 2^7

This rule also applies to expressions with negative exponents and fractional exponents. For example, if you have (1/2)^3 * (1/2)^(-2), you can use the product rule for exponents to simplify it as:

(1/2)^3 * (1/2)^(-2) = (1/2)^(3-2) = 1/2

Note that the product rule for exponents only applies when the bases are the same. If the bases are different, you cannot use this rule.

It is also important to note that the product rule does not apply to zero exponents. A zero exponent means that the base is raised to the power of zero, which is always equal to one. Therefore, the product rule cannot be used when one of the exponents is zero.

In summary, the product rule for exponents is a powerful tool for simplifying expressions with the same base. By adding the exponents, you can quickly and easily simplify complex expressions with negative exponents, fractional exponents, and positive exponents.

### Factor and Undefined

When applying the product rule for exponents, it is important to understand the concept of factoring. Factoring is the process of breaking down a number or expression into its constituent parts. For example, the number 12 can be factored into 2 x 2 x 3, while the expression x^2 + 3x + 2 can be factored into (x + 1)(x + 2).

In terms of the product rule for exponents, factoring is useful when dealing with expressions that have multiple terms. For example, consider the expression (xy)^3. Applying the product rule, we get (xy)^3 = x^3 y^3. However, we can also factor the expression as (xy)^3 = (x^3)(y^3). This can be useful when simplifying more complex expressions.

Another important concept to understand when working with exponents is undefined values. An undefined value is a value that is not defined or does not exist in the context of the problem. For example, consider the expression 0^0. This expression is undefined because any number raised to the power of 0 is equal to 1, except for 0^0, which has no defined value.

In addition to undefined values, it is also important to understand the concept of zero exponents. A zero exponent is an exponent of 0, which means that the base is raised to the power of 0. Any number raised to the power of 0 is equal to 1, except for 0^0, which, as mentioned earlier, is undefined. For example, 2^0 = 1, 10^0 = 1, but 0^0 is undefined.

Understanding these concepts is crucial when working with exponents and applying the product rule. By factoring expressions and avoiding undefined values and zero exponents, one can simplify expressions and arrive at the correct solution.

## Product Rule for Exponents FAQ

### What is the rule for multiplying exponents?

The rule for multiplying exponents is to add the exponents when the bases are the same. For example, if you have x^2 * x^3, you can add the exponents to get x^(2+3) = x^5.

### How do you use the product rule for exponents?

To use the product rule for exponents, you need to multiply the bases and add the exponents. For example, if you have (x^2)(x^3), you can multiply the bases to get x^(2+3) = x^5.

### What is the product rule of exponents in algebra?

The product rule of exponents in algebra is a rule that allows you to simplify expressions that involve multiplication of exponential terms. The rule states that when you multiply two exponential terms with the same base, you can add the exponents.

### What are the basic exponent rules?

The basic exponent rules are the product rule, quotient rule, power rule, negative exponent rule, and zero exponent rule. These rules help simplify expressions with exponents.

### What is the rule for multiplying exponents with different bases?

When multiplying exponents with different bases, you cannot simplify the expression unless the bases are the same.

### What is the rule for multiplying exponents with the same base?

The rule for multiplying exponents with the same base is to add the exponents. For example, x^2 * x^3 = x^(2+3) = x^5.

### Do you multiply or do exponents first?

When you have an expression with both multiplication and exponents, you should perform the exponentiation first, and then perform the multiplication.

### Do you calculate exponents before multiplying?

Yes, you should calculate exponents before multiplying. This is because exponents have a higher priority than multiplication in the order of operations.

## Product Rule for Exponents Worksheet Video Explanation

Watch our free video on how to Multiply Exponents. This video shows how to solve problems that are on our free Product Rule Exponents worksheet that you can get by submitting your email above.

Watch the free video on How to Multiply Exponents on YouTube here: Product Rule for Exponents

Video Transcript:

This video is about exponent product rule​. You can download our product rule for exponents worksheet by clicking on the link in the description below.

Here we are at number one. Number one says we’re going to multiply 8 to the 4th times 8 to the 11th. In terms of this problem when we have 8 to the 4th, what that really is saying is 8 times 8 times 8 times 8, and then we have 8 to the 11th. What 8th to the 11th is saying is we’re multiplying 8 to the 4th times 8 to the 11th or we have 11 8’s. If you look at our problem we have 8 to the 4th, there are 4 8s and then we have multiplied times 8 to the 11th, which are 11 8s. When you multiply all those together you have to figure out how many eights you are going to end up with. What we’re going to do is we’re going to count 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8. Our final answer will be 8 to the 15th power.

You can skip this step if you know the product rule exponents. You will notice that what we did was we counted up all of the 8 but instead of having to do that you could have just added the exponents. If you look back to our original problem of 8 to the 4th times 8 to the 11th. If you just do 4 plus 11 you will get the same answer 8 to the 15th. The shortcut for using the product rule for exponents is to just add the exponents together as long as they have the same base.

Our next example gives us 4 to the 8th times the four to the fifth eight to the third. Now we learned in our first example the rules for multiplying exponents can be just he add the exponents. What we’re going to do is we’re going to take the fours and we’re going to add the exponents together and then we’re going to take the base of eight and rewrite it underneath. Now the reason we don’t write the two together is because the bases are different.

This is a base of four and they can get multiplied or combine together and this has a base of eight and it cannot go with the fours. If we had hypothetically another eight here we could have multiplied the aides together but we don’t have another eight. We just leave the eight by itself when using the shortcut we’re going to add the exponents to the four. We will do four to the eight plus five which is four to the 13th power and then we have this eight to the third that is getting combined with nothing else. It will just stay 8 to the third and that is our answer.

The final example that we’re going to go over shows when we have a negative exponent. Just because we have a negative exponent does not mean the product rule of exponents changes. We still have a base of seven. We know that because the base of seven for both of these we’re going to add them together. We’re going to keep the base and then we’re going to add negative three plus the exponent of 17. When you add negative 3 plus 17 you get 14 and that’s gonna be your answer. The key takeaway here is that just because we have a negative exponent does not change the rule, the rules stay the same. You can try all the practice problems by downloading the free product rule of exponents worksheet above.