Oexponent rulesExplain how to solve several equations that, as expected, contain exponents. But there are different types of exponential equations and exponential expressions that can seem daunting at first.
Mastering these basic exponent rules along with basic logarithm rules (also known as "logarithm rules") will make your study of algebra very productive and enjoyable. Note that during this process, the order of operations still applies.
Like most math tactics, they existteaching strategiesYou can use it to follow exponent rules easily.
To help you teach these concepts, we have aFree Exponent Rules Worksheet for you to download and use in your class!
What are exponents?
Exponents, also called powers, are values that indicate how many times a base number must be multiplied by itself. For example, 43 tells you to multiplyfourby yourselfthreemal.
43= 4 × 4 × 4 = 64
A number raised by a power is called a datum.Base, while the superscript above is theExponentorPerformance.
Credit: For the square inch
The above equation is called "four to the power of three". The power of two can also be said as "checked' and the power of three can be said as 'rolled“. These terms are often used to find area or volume in various ways.
Writing a number in exponential form refers to simplifying it to a power base. For example rotate5 × 5 × 5look in exponential form53.
Exponents are a way to simplify equations to make them easier to read. This becomes especially important when dealing with variables like “𝒙” and “𝑦” – like𝒙7× 𝑦5= ?it's easier to read than(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝑦)(𝑦)(𝑦)(𝑦)(𝑦) = ?
Rules of exponents in everyday life
Understanding the properties of exponents will not only help you solve various algebraic problems, exponents are also used practically in everyday life when calculating square feet, square meters and even cubic centimeters.
Exponent rules also simplify the calculation of extremely large or extremely small quantities. They are also used in the computing and technology world when describing megabytes, gigabytes and terabytes.
What are the different exponent rules?
There isSevenRules of exponents or laws of exponents your students need to learn. Each rule shows how to solve different types of math equations and how to add, subtract, multiply, and divide exponents.
Be sure to go through each exponent rule in class, as each plays an important role in solving exponent-based equations.
1. Power product rule
When multiplying two bases by the same value, keep the bases the same and add the exponents to get the solution.
42× 45= ?
Since the base values are four, keep them the same and add the exponents (2 + 5) together.
42× 45= 47
Then multiply four by itself seven times to get the answer.
47= 4 × 4 × 4 × 4 × 4 × 4 × 4 = 16,384
Let's expand on the above equation to see how this rule works:
In an equation like this, adding the exponents is a shortcut to getting the answer.
Here's a trickier question to try:
(4𝒙2))(2𝒙3) = ?
Multiply the coefficients together (four and two) as they don't have the same base. Then keep the '𝒙' and add the exponents.
(4𝒙2))(2𝒙3) = 8𝒙5
2. Power Quotient Rule
Multiplication and division are opposites - the quotient rule acts as the opposite of the product rule.
When dividing two bases with the same value, keep the base the same and subtract the exponent values.
55÷ 53= ?
Both bases in this equation are five, which means they stay the same. Then take the exponents and subtract the divisor from the dividend.
55÷ 53= 52
Finally, simplify the equation if necessary:
52= 5 × 5 = 25
Again, expanding the equation shows us that this shortcut gives the correct answer:
See this more complicated example:
5𝒙4/ 10𝒙2= ?
The same variables in the denominator cancel those in the numerator. You can show this to your students by crossing out the same number of 𝒙 at the top and bottom of the fraction.
5𝒙4/ 10𝒙2= 5𝒙/10
Then simplify whenever possible, as you would any fraction. Five can become ten, five times the fraction with the remaining 𝒙 variables becomes ½.
5𝒙4/10𝒙2= 1𝒙2/2 = 𝒙2/2
3. Power of a power rule
This rule shows how to solve equations involving a powerbehavedof another power.
(𝒙3)3= ?
In equations like the previous one, multiply the exponents and keep the base the same.
(𝒙3)3= 𝒙9
Check out the expanded equation to see how this works:
4. Power of a product rule
If any base is multiplied by an exponent,to distributethe exponent forall partsa base.
(𝒙𝑦)3= ?
In this equation, the power of three must be distributed over the variables 𝒙 and 𝑦.
(𝒙𝑦)3= 𝒙3𝑦3
This rule applies when exponents are also connected to the base.
(𝒙2𝑦2)3= 𝒙6𝑦6
Expanded, the equation would look like this:
Both variables arecheckedin this equation and arebehavedthree tall. This means that three is multiplied by the exponents on both variables, making them variables raised to the power of six.
5. Power of a quotient rule
A quotient simply means that you are dividing two quantities. In this rule you areincrease a quotientby a force. Like the power of a product rule, the exponent must be divided between all values within the parentheses to which it is attached.
(𝒙/𝑦)4= ?
Here, increase both variables inside the square brackets by a power of four.
Take this more complicated equation:
(4𝒙3/5𝑦4)2= ?
Don't forget to distribute the exponent you are multiplying byboththe coefficient and the variable. Then simplify where possible.
(4𝒙3/5𝑦4)2= 42𝒙6/52𝑦8= 16𝒙6/25𝑦8
6. Zero power rule
Any base raised to the power of zero equals one.
The easiest way to explain this rule is to use the power quotient rule.
43/43= ?
Using the power quotient rule, subtract the exponents from each other, which cancels them out and leaves only the base. Every number divided by itself is one.
43/43= 4/4 = 1
No matter how big the equation is, anything raised to the power of zero becomes one.
(82𝒙4𝑦6)0= ?
Normally, the outer exponent would have to be multiplied by each number and variable in parentheses. However, because this equation is raised to the zero power, these steps can be skipped and the answer simply becomes one.
(82𝒙4𝑦6)0= 1
The fully expanded equation would look like this:
(82𝒙4𝑦6)0= 80𝒙0𝑦0= (1)(1)(1) = 1
7. Negative Exponentenregel
If a number is increased by a negative exponent, transform it into a reciprocal to change the exponent to a positive one.NoUse the negative exponent to transform the base to negative.
Photo credit: Thinglink
We already talked about reciprocity in our article: “How to divide fractions in 3 easy steps“. Essentially, reciprocals are what you multiply a number to get the value of one. For example, to make two one, multiply by ½.
Now look at this exponent example:
𝒙-2= ?
To convert a number to a reciprocal:
- Convert the number to a fraction (place it over one)
- Convert numerator to denominator and vice versa
- When a negative number switches places in a fraction, it becomes a positive number.
The purpose of equations with negative exponents is to make them positive.
Now look at this more complicated equation:
4𝒙-3𝑦2/20𝒙𝑧-3= ?
In this equation there are two exponents with negative powers. Simplify what you can, then invert the negative exponents to their reciprocal form. In the solution, 𝒙-3 goes to the denominator while 𝑧-3 goes to the numerator.
Since there is already a value 𝒙 in the denominator, 𝒙3 is added to that value.
4𝒙-3𝑦2/20𝒙z-3= 𝑦2𝑧3/5𝒙4
With these seven rules in your students' back pocket, they'll be able to answer most of the exponential questions they come across!
Exponent rules diagram
How Prodigy can help you teach exponent rules
Prodigy is a curriculum-aligned math game that lets you assign questions, track progress, and identify pain points in your students' learning. And you can create teacher and student accounts for free!
With so many different exponent rules to follow and multiple students to keep track of, it can be difficult to identify who needs help with what. Prodigy makes it easy to track progress and create a unique gaming experience for each student based on their needs.
Stats are tracked live as students play and feedback is immediately available. Most of the time, your students don't even realize they're in math class. It's all part of your personalized gaming experience!
From the teacher's dashboard, you can create lesson plans, view live statistics, enter custom assignments and prepare your students for upcoming tests.See how you can use Prodigy:
- Prepare students for standardized tests
- Reinforce concepts in the classroom(like exponent rules)
- differentiate math practicein math class and at home
Free Exponent Rules Worksheet
Math worksheets are useful tools that can show students how to understand key concepts. You can see how students are finding answers, where they are having problems, and if concepts need to be explored in more detail.
We, with the help of our teaching staff, have created an exponent rules worksheet to help you teach exponents.
Click here to download our Exponent Rules Worksheet, complete with an answer key!
Conclusion: Practice the exponent rules
Exponents are used to indicate how many times a base value is multiplied by itself. This simplifies the equations into an easier to read format. (𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙)(𝑦𝑦𝑦𝑦𝑦𝑦)(𝑧𝑧𝑧𝑧𝑧) = 𝒙9𝑦6𝑧5
As a reminder, there are seven basic rules that explain how to solve most mathematical equations involving exponents. The exponent rules are:
- Power Product Rule - Add powers when multiplying like bases
- power quotient rule— Subtract powers when dividing like bases
- power of power reigns— Multiply powers by increasing a power by another exponent
- power of a product rule- Distribute power to each base by increasing multiple variables by one power
- Power of the Quotient Rule– Distribute the power among all values in a quotient
- zero power rule—Any base raised to the power of zero becomes a
- Negative exponentenregel—To change a negative exponent to a positive one, rotate it to a reciprocal
Exponents tend to appear throughout our lives, so it is important for students to understand how they progress. There are many rules to remember, but once your students understand them, solving exponents will likely become easier!