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:

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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.

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:

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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 Groundhog Day Math Puzzle for Grades 3-8

Happy Groundhog Day!

Is it just me or is Groundhog Day not as big of a deal as it used to be?

Each year we celebrate Groundhog Day on February 2nd and anxiously await Punxsutawney Phil’s prediction of how much longer winter will last.

And while his track record is pretty poor (his predictions are only correct 40% of the time according to the National Oceanic and Atmospheric Administration), you can still cling to hope for a shorter winter and celebrate in your classroom this year with a fun Groundhog Day Math Puzzle for students in grades 3-8!

The activity can be used a warm-up, transition, homework assignment, or exit ticket and can be shared with students in grades 3 and up!

Image Source: Mashup Math MJ

Groundhog Day Puzzle #1

Keep reading to see the answer key.

Hint: Many people have zero confidence in the groundhog's prediction for early spring

Answer Key: Groundhog=12, Sun=0, Hat=6, ?=6

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Groundhog Day Puzzle #2

Answer Key: Cloud=7, Snowflake=20, ?=33 (the groundhog and the hat can be any two numbers with a sum of 13)

Are YOU looking for some super fun daily math puzzles for your students in grades 3-8? Our best-selling 101 Daily Math Challenges workbook is now available as a PDF download!

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How will you use these math puzzles with your kids? Share your thoughts and suggestions in the comments section 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 and an advisor to Amazon Education's 'With Math I Can' Campaign. 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 Math Coloring Pages and Worksheets for Grades 1-8

Are you looking for fun, engaging, and printable Math Coloring Pages and Math Coloring Worksheets for students in grades 1-8?

If so, then go ahead and share the following math coloring activities in your upcoming lesson plans and let the fun begin! The activities make perfect warm-up, cool-down, and/or transition activities.

Wait! Do you want free K-12 math resources in your inbox every week? Sign-up for our free email newsletter and start getting free stuff today!

Tags: math coloring pages, math coloring worksheets, 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade, 6th grade, 7th grade, 8th grade, grade 1, grade 2, grade 3, grade 4, grade 5, grade 6, grade 7, grade 8, first, second, third, fourth, fifth, sixth, seventh, eighth

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Ice Cream Math Coloring Page 01

Keep reading to see the answer key

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Ice Cream Math Coloring Page 02

Keep reading to see the answer key

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Ice Cream Math Coloring Page 03

Keep reading to see the answer key

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Are you looking for more Math Coloring Activities to share with your kids?

My best-selling workbook 101 Math Puzzles for Kids Coloring Book for Ages 8+ is now available as a PDF download. You can get yours today by clicking here.

And click here to access sample pages from the book.

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Answer Key

Puzzle #1: Strawberry=9, Cone=5, Sundae=4, ?=18

Puzzle #2: Strawberry=8, Cone=21, Sundae=24, ?=53

Puzzle #3: Strawberry=14, Cone=7, Sundae=6, ?=56

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Are You Ready for 17 Awesome New Math Challenges?

17 Fun and Printable Math Puzzles for Elementary and Middle School Students

A Post By: Anthony Persico

Winter vacation is over and teachers are taking on the challenge of engaging students and getting them refocused and thinking mathematically again.

Many teachers utilize engaging activities to kickstart the new year and get their kids excited about learning math.

So, in the spirit of getting students excited (and feeling challenged), I have created the following collection of free New Years Math Challenges for you to share with your kids!

Each printable math worksheet shares a challenge question (answers are included) appropriate for all elementary and middle school grade levels.

Parents can also share these fun challenge questions with your kids to keep them engaged and thinking mathematically outside of school.

Now let's get problem-solving!

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1.) Day 1 of 17

(Hint: Start with the chicks!)

Koala Bear = 7

Camel = 3

Chick = 11

Gorilla = 4

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2.) Day 2 of 17

(Hint: Start with the grapes!)

Pineapple = 4

Strawberry = 12

Grape = 12

Watermelon = 2

? = 26

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3.) Day 3 of 17

(Hint: The racket and the arrow both represent the same value!)

Bowling = 50

Basketball = 25

Arrow = 5

Racket = 5

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4.) Day 4 of 17

(Hint: Start with the blue cars!)

Blue Car = 9

Scooter = 7

Helicopter = 8

Jet = 4

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5.) Day 5 of 17

(Hint: 8 minus what number is that same number?)

T.V. = 4

Camera = 8

iPhone = 4

Microscope = 6

? = 6

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6.) Day 6 of 17

(Hint: You have to crawl before you can walk!)

Baby = 6

Hard Hat = 1

Old Man = 2

Police Officer = 6

The baby and the police officer both represent a value of 6. They are interchangeable. 

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Are you looking for more daily math challenges and puzzles to share with your kids?

Our best-selling workbook 101 Math Challenges for Engaging Your Students in Grades 1-8 is now available as a PDF download!

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7.) Day 7 of 17

(Hint: Equivalent emojis are interchangeable!)

Yellow Face = 10

Green Face = 10

Red Face = 17

? = 17

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8.) Day 8 of 17

(Hint: Even if you can't find the snail and the alligator individually, you know what their sum is.)

Turtle = 5

Alligator = 7 and Snail = 0, or

Snail = 7 and Alligator = 0

? = 12

One of the pair (alligator and snail) is 7 and the other is 0, but it is not possible to tell which is which.

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9.) Day 9 of 17

(Hint: Start with the best food--pizza!)

Cheeseburger = 2

Taco = 8

Pizza = 3

? = 24

If the value of the cheeseburger was zero, then the value of the pizza and the taco would also become zero.

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10.) Day 10 of 17

(Hint: Any nonzero value divided by itself equals one!)

Saxophone = 4

Guitar = 4

Keyboard = 4

? = 12

The value of the guitar and the value of the saxophone must be equal (and not zero), since their quotient is one! 

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11.) Day 11 of 17

(Hint: Handle that volcano right away!)

Volcano = 1

Statue of Liberty = 3

Rocket = 6

Race Car = 2

Ferris Wheel = 18

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12.) Day 12 of 17

(Hint: Patterns, my friend...patterns!)

Pig = 2

Tiger = 3

Mouse = 4

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 “I always look forward to getting my Mashup Math newsletter email every week. I love the free activities!” -Christina R., 5th Grade Math Teacher, Dallas, TX

Do YOU want free math resources, lesson activities, and puzzles and games for grades 1-8 in your inbox every week? Join our mailing list and start getting tons of free stuff (including a free PDF Math Workbook)!

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13.) Day 13 of 17

(Hint: Think about what happens when you multiply values by zero and by one)

Avocado = 1

Pancakes = 4

Banana = 6

Donut = 8

Coffee = 0

French Fries = 16

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14.) Day 14 of 17

(Hint: The whole diagram represents 182 and the two given sections represent 112. How can you distribute the difference of 70 between the two unknown sections?)

Clown Fish = 4

Octopus = 10

Shark = 40

Crab = 30

Whale = 3

Area Model: 14 x 13

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15.) Day 15 of 17

(Hint: This is like a hundreds chart...kinda)

Neon Green = 510

Neon Pink = 80

Neon Blue = 220

Neon Orange = 190

Each box equals 10.

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16.) Day 16 of 17

(Hint: Look for patterns within the patterns!)

Case 1: Increases by 2, then 3, then 4...

? = 17

Case 2: Each value is twice the previous value

? = 48

Case 3: Each value is one more than thrice the previous value 

? = 283

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17.) Day 17 of 17

(Hint: Just play around with this one until you get it!)

Cow = 3

Fox & Boar = 6 and 7

Monkey & Frog = 8 and 4

Brown Bear & Panda = 9 and 5

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Are you looking for more daily math challenges and puzzles to share with your kids?

Our best-selling workbook 101 Math Challenges for Engaging Your Students in Grades 1-8 is now available as a PDF download!

Do you and your kids love our daily math challenges and puzzles? Have thoughts? Please share them in the comments below!

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Read More Posts About Math Education:

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math and an advisor to Amazon Education's 'With Math I Can' Campaign. 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.