Free Guide: Multiplying Exponents Explained

Multiplying Exponents: The Complete Guide

Are you ready to master the laws of exponents and learn how to Multiply Exponents with the Same Base with ease?

This lesson is part of our Rules of Exponents Series, which also includes the following lesson guides:

• Dividing Exponents

• Working with Fractional Exponents

• Working with Negative Exponents

Let’s start with the following key question about multiplying exponents:

•  How can you multiply powers (or exponents) with the same base?

• Note that the following method for multiplying powers works when the base is either a number or a variable (the following lesson guide will show examples of both)

Multiplying Exponents Example

Let’s start with a simple example: what is 3^3 times by 3^2?

*Notice that each term has the same base, which, in this case is 3.

Start by rewriting each term in expanded form as follows (you won’t have to do this every time, but we’ll do it now to help you understand the rule, which we’ll get to later.)

Since we have 3 being multiplied by itself 5 times ( 3 x 3 x 3 x 3 x 3 ), we can say that the expanded expression is equal to 3^5

And we can conclude that: 3^3 x 3^2 = 3^5

But Why?

Do you notice a relationship between the exponents?

Multiplying Exponents Rule

Did you notice a relationship between all of the exponents in the example above?

Notice that 3^2 multiplied by 3^3 equals 3^5. Also notice that 2 + 3 = 5

This relationship applies to multiply exponents with the same base whether the base is a number or a variable:

Whenever you multiply two or more exponents with the same base, you can simplify by adding the value of the exponents:

Here are a few examples applying the multiplying exponents rule:

Multiplying Exponents Rule Example (Variable Base)

Example: Simplify (X^5)(X^7) ?

Solution: (X^5) (X^7) = X^12 …because 5 + 7 = 12

Multiplying Exponents Rule Example (Numerical Base)

Example: Simplify (8^3)(8^5) ?

Solution: (8^3) (8^5) = 8^8 …because 3 + 5 = 8

Multiplying Powers with the same base is as simple as that!

If you still need help, check out this free Multiplying Exponents video lesson:

Video: Multiplying Exponents Explained in 3 Minutes!

Multiplying Exponents Practice Worksheet and Answer Key

Are you looking for some extra multiplying exponents practice?

Click the link below to download your free Multiplying Exponents Worksheet (PDF) and Answer Key!

>>> Click here to get your free Multiplying Exponents Worksheet.

(answer key included)

Ready to Keep Learning? Here are More Free Related Math Resources:

• How to Convert a Fraction to a Decimal

• Dividing Exponents: The Complete Guide

• Negative Exponent Rule Explained in 3 Easy Steps

• Video Lesson: Scientific Notation Explained

• The Ultimate Guide to Cube Roots

• Free PDF Math Worksheets (with Answers)

• Activity: Here’s an Awesome Way to Teach Kids Fractions

Keep Learning:

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Share your ideas, questions, and comments below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math . You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

Free Guide: Dividing Exponents Explained

Dividing Exponents: Everything You Need to Know

Are you ready to become a master of dividing exponents?

(Need help with Negative Exponents, click here for our super easy 3-step explanation)

Let’s start with the following key question about dividing exponents:

•  How can you divide powers (or exponents) with the same base?

• Note that the following method works when the base is either a number or a variable (the following lesson guide will show examples of both)

Dividing Exponents Example

Let’s start with a simple example: what is 5^7 divided by 5^4?

Start by rewriting the fraction in expanded form as follows (you won’t have to do this every time, but we’ll do it now to help you understand the rule, which we’ll get to later.)

Notice how there are repeat values in the numerator and the denominator.

Since any non-zero value divided by itself is equal to one, you can cancel out these repeats as follows:

Doing this allows you to rewrite the above fraction in simplified form, as follows:

Finally, you can rewrite 5 x 5 x 5 as 5^3 and you have finished the problem.

5^7 divided by 5^4 equals 5^3

Dividing Exponents Rule

Did you notice a relationship between all of the exponents in the example above?

Notice that 5^7 divided by 5^4 equals 5^3. Also notice that 7 - 4 = 3

This relationship applies to dividing exponents with the same base whether the base is a number or a variable:

Whenever you divide two exponents with the same base, you can simplify by subtracting the value of the exponent in the denominator by the value of the exponent in the numerator.

Here are a few examples applying the rule:

Dividing Exponents Rule Example (Variable Base)

Dividing Exponents Rule Example (Numerical Base)

Dividing exponents with the same base is as simple as that!

If you still need help, check out this free Dividing Exponents video lesson:

Video: Dividing Exponents Explained in 3 Minutes!

Dividing Exponents Practice Worksheet and Answer Key

Are you looking for some extra dividing exponents practice?

Click the link below to download your free Dividing Exponents PDF Worksheet and Answer Key!

>>> Click here to get your free Dividing Exponents Worksheet.

(answer key included)

Ready to Keep Learning? Here are More Free Related Math Resources:

• How to Convert a Fraction to a Decimal

• Negative Exponent Rule Explained in 3 Easy Steps

• Video Lesson: Scientific Notation Explained

• The Ultimate Guide to Cube Roots

• Free PDF Math Worksheets (with Answers)

• Activity: Here’s an Awesome Way to Teach Kids Fractions

Keep Learning:

---

Share your ideas, questions, and comments below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math . You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

Fractional Exponents Explained! (Free Worksheet Included)

Fractional Exponents: Everything You Need to Know

Are you ready to learn how to work with Fractional Exponents?

(Need help with Negative Exponents, click here for our super easy 3-step explanation)

Before you learn how to work with fractional exponents and use them to express powers and roots together, let's do a quick vocabulary review:

•  It is important that you are familiar with identifying the root index and radicand before moving forward.

• Note: If a radical symbol does not include a root index value, the number 2 is implied (square root).

• Example:

The radicand is 64 and the root index is 3. The left side of the equation implies “the cube root of 64”

What is a Fractional Exponent?

Now that you are able to identify the root index and the radicand, you are ready to understand fraction exponents.

A fractional exponent is another way of expressing powers and roots together.

You will now learn how to express a value either in radical form or as a value with a fractional exponent.

Example: Express the square root of 49 as a fractional exponent.

Note that we can rewrite 49 as 49^1 since they are equivalent (you will see in a minute why this is helpful)

In this case, the square root of 49 is equal to 49^(1/2) because they are both equal to 7.

But why?

Why is the square root of 49 equal to 49^(1/2)?

Fractional Exponents Rule

Here is the rule that you can use to work with fractional exponents from now on:

Now, let’s apply the above rule to two more examples:

Example A:

Example B:

Fractional Exponents Calculator

If you need a fast and easy way to convert between radical expressions and expressions with fractional exponents, then you can take advantage of the many free online fractional exponents calculators that are available.

This free fractional exponents calculator from www.calculatorsoup.com shares all of the steps involved in converting and also simplifies.

To use the fractional exponent calculator, simply input the base value, the value of the numerator and the value of the denominator and press calculate.

Are you looking to learn more about working with decimals and understanding place value? Click here for more free resources

Fractional Exponents Worksheet

Do you need more practice with learning how to work with fractional exponents? The following fractional decimals worksheet and answer key will give you plenty of opportunities to practice.

Click here to download your free Fractional Exponents Worksheet and Answers Key.

(answer key included)

And if you are looking for a more in-depth lesson on how to work with fractional exponents? Check out this awesome Fractional Exponents Explained! video lesson:

More Free Related Math Resources:

• How to Convert a Fraction to a Decimal

• Negative Exponent Rule Explained in 3 Easy Steps

• Video Lesson: Scientific Notation Explained

• The Ultimate Guide to Cube Roots

• Free PDF Math Worksheets (with Answers)

• Activity: Here’s an Awesome Way to Teach Kids Fractions

Keep Learning:

---

Share your ideas, questions, and comments below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math . You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.