About This Chapter
Standard: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)3 = 5( to the 1/3)3 to hold, so (5 to the 1/3)3 must equal 5. (CCSS.Math.Content.HSN-RN.A.1)
Standard: Rewrite expressions involving radicals and rational exponents using the properties of exponents. (CCSS.Math.Content.HSN-RN.A.2)
Standard: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. (CCSS.Math.Content.HSN-RN.B.3)
About This Chapter
A clear understanding of rational exponents enables students to identify rational and radical numbers as well as solve arithmetic problems using rational and irrational numbers. Students who comprehend the elements of rational exponents will be able to simplify expressions and find the distributive properties of expressions using rational exponents.
The lessons in this standard cover the following topics:
- The five main exponent properties
- Zero and the negative exponent
- Simplifying expressions with exponents
- Different types of numbers
- Rationalizing the denominator
- Addition, subtraction and multiplication with rational exponents
- Dividing expressions with rational exponents
- The distributive property and expressions with rational exponents
- Sums and products of rational and irrational numbers
Students show they've mastered this standard by being able to correctly identify the properties of rational exponents and perform mathematical equations based on the types of numbers used in the expressions. This standard and these lessons can prepare them for college and careers, such as in financial, construction and engineering industries.
How to Use These Lessons in Your Classroom
The following are some tips on how you can incorporate these lessons on rational exponents into your curriculum to help meet the common core standards:
Exponent Identification Lessons
Have students watch the video lessons that cover the different types of numbers, radical and rational exponents and the main properties of exponents in the classroom or as homework assignments. Create two or three rewritten rational exponents, and pass one out to each student. Have the students group themselves based on whether their assigned rational exponents are the same.
Simplifying Expressions Lessons
Watch the Simplifying Expressions with Rational Exponents lesson in the classroom or assign it for homework. Go over the practice lesson on the same topic together in class.
Pre-Quiz and Post-Quiz Lessons
Have students take the quizzes for the Multiply and Divide Expressions with Rational Exponents and Add and Subtract Expressions with Rational Exponents to introduce the concepts of arithmetic problems using rational expressions. Watch the video lessons, then have students re-take the quizzes to assess their comprehension.
1. What Are the Five Main Exponent Properties?
We'll look at the five important exponent properties and an example of each. You can think of them as the order of operations for exponents. Learn how to handle math problems with exponents here!
2. How to Define a Zero and Negative Exponent
The zero and negative exponent properties are two you will use quite a lot in mathematics. The negative exponent property can be confusing, but when you remember a couple fun ideas, you will get it right every time!
3. How to Simplify Expressions with Exponents
In this video, we will put all of the exponent properties we have learned together. Don't let lots of numbers and letters confuse you! Simplifying exponents can be easy and fun!
4. What are the Different Types of Numbers?
Learn all of the different types of numbers: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. This video lesson teaches you how to classify any numbers you come across.
5. How to Find the Square Root of a Number
What is a square root? In this lesson, we'll learn what a square root is and how to find it. We'll review a variety of examples in order to master the concept.
6. Radicands and Radical Expressions
It's easy to see that funky radical symbol and panic just a little bit. There are so many rules that go along with it, it's hard to keep up with them all. This lesson will help by describing what you should do when you are faced with a square root.
7. Evaluating Square Roots of Perfect Squares
Squares and square roots are inverse, or opposite, operations involving radicals. Learn how to determine the square root of perfect squares in this lesson.
8. Factoring Radical Expressions
Watch this video lesson to learn how to apply the product rule to your radical expressions. Learn how this product rule actually helps you to simplify your radicals as well.
9. Simplifying Square Roots of Powers in Radical Expressions
Simplifying radical expressions that contain powers can be tricky. There are a few simple rules that will help you perform these simplifications with ease. This lesson will teach you how.
10. Multiplying then Simplifying Radical Expressions
Multiplying 2 or more radical expressions uses the same principles as multiplying polynomials, with a few extra rules for dealing with the radicals. This lesson will teach you how to multiply and then simplify radical expressions.
11. Dividing Radical Expressions
When dividing radical expressions, we use the quotient rule. This lesson will describe the quotient rule and how to use it to solve these radical expressions.
12. Simplify Square Roots of Quotients
The quotient rule can be used to simplify square roots of quotients. This lesson will define the quotient rule and show you how it is used to simplify square roots.
13. Rationalizing Denominators in Radical Expressions
Radical expressions containing denominators are not simplified completely unless the denominator is free of radical symbols. This lesson will teach you how to remove a radical from the denominator of a fraction through a process called rationalizing the denominator.
14. Addition and Subtraction Using Radical Notation
There are specific rules governing adding and subtracting radical expressions. This lesson will describe these rules and give examples of how they are used.
15. Multiplying Radical Expressions with Two or More Terms
Multiplying radical expressions with more than two terms can be confusing. This lesson will take some of the confusion away by giving clear steps for multiplying these expressions. It will also provide some examples to help solidify the steps.
16. Solving Radical Equations: Steps and Examples
Solving radical equations is not any more difficult than solving other algebraic equations. This lesson will show you how to solve equations containing a square root and give some real-world examples.
17. Solving Radical Equations with Two Radical Terms
Solving equations with two radical terms takes some patience and care, but it really is not difficult. This lesson will show you the steps to solve these more complicated equations plus give you some examples to follow.
18. Simplifying Expressions with Rational Exponents
Simplifying expressions with rational exponents is so easy. In fact, you already know how to do it! We simply use the exponent properties but with fractions as the exponent!
19. Rational Exponents
In this video, learn how to go from a rational exponent to a radical expression and back. No tricks or magic, just good math! We'll review the basics and look at a few examples.
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