# Ch 1: Common Core HS Math: Rational Exponents

### 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?

There are five main exponent properties, which are much like the order of operations in exponents, that give structure to simplifying expressions. Learn the formulas of the five exponent properties, including product of powers, power to a power, quotient of powers, power of a product, and power of a quotient.

### 2. How to Define a Zero and Negative Exponent

Exponents are not always simple integers and can take the form of negative numbers or even zero. Learn to define zero and negative exponents, explore the differences in how a negative number functions as an exponent, and understand the properties of zero and negative exponents through examples.

### 3. How to Simplify Expressions with Exponents

Expressions can be rewritten using exponents to be simplified visually and mathematically. Explore the use of several properties used to simplify expressions with exponents, including the product of powers, power to a power, quotient of powers, power of a product, and the zero property.

### 4. What are the Different Types of Numbers?

There are different types or families of numbers. Learn how to identify natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

### 5. How to Find the Square Root of a Number

Calculating the square root of a number is the inverse operation of squaring that number. Explore how to find square roots, the square of numbers, perfect and imperfect squares, and some square root practice examples.

### 6. Radicands and Radical Expressions

Radical expressions refer to expressions that include radicals, or square root signs. Explore the definition of a radical symbol and radicand, and understand the rules for radical expressions, including why they only work with real numbers.

### 7. Evaluating Square Roots of Perfect Squares

A Perfect Square is found when a number is the square of another integer. Learn the notation of radicals in the language of math, and how to calculate the square-roots of Perfect Squares.

### 8. Factoring Radical Expressions

Radical expressions can be factored by following the product rule for radicals. Learn about radical expressions, how to identify and factor them, and how to apply the product rule to simplify a radical expression.

### 9. Simplifying Square Roots of Powers in Radical Expressions

Radical expressions are used to represent numbers that can be reduced to their base number based on their exponent. Learn about the description of a radical symbol, radical expressions with exponents, and how to simplify radical expressions.

### 10. Multiplying then Simplifying Radical Expressions

The terms under radical symbols with the same index can be combined through multiplication, then simplified. Learn how to multiply and simplify radical expressions by using the product rule, and review the examples.

### 11. Dividing Radical Expressions

Dividing of radical expressions is possible through using the quotient rule. Discover more about dividing radical expressions, including simplification of expressions and rationalization of the denominator.

### 12. Simplify Square Roots of Quotients

The quotient rule can be used to solve radical expressions. This lesson explores this idea and provides examples demonstrating how to simplify the square roots of quotients.

### 13. Rationalizing Denominators in Radical Expressions

Rationalizing denominators is a process that is used to simplify radical expressions as much as possible. Discover when a radical expression is finished, and learn how to rationalize a denominator with one term or more than one term.

### 14. Addition and Subtraction Using Radical Notation

In mathematics, a radical notation indicates the square root of a number. Learn how to do addition and subtraction using radical notation, and recognize that mathematics and algebra include terms that may not be like others though radicals are regarded as like terms.

### 15. Multiplying Radical Expressions with Two or More Terms

Multiplying radical expressions with two or more terms involves using the distributive property. Learn about multiplying radical expressions, the distributive property, and multiplying two binomials.

### 16. Solving Radical Equations: Steps and Examples

A radical equation is an equation in which a variable is under a radical. Learn to identify radicals, perform inverse operations, and solve radical equations through real-world examples.

### 17. Solving Radical Equations with Two Radical Terms

An equation containing a square root symbol, or radical, is called a radical equation. To solve a radical equation, you isolate the radical by squaring it. Learn the step-by-step process on how to solve radical equations with two radical terms by using the FOIL method and checking your work.

### 18. Simplifying Expressions with Rational Exponents

Apart from being expressed as fractions, rational exponents function the same as other exponents when the properties of exponents are applied to them. Follow the steps in a series of examples to see how expressions with rational exponents are simplified in each context.

### 19. Rational Exponents

Rational exponents are simply a fraction representing an exponent that is rewritten as a radical expression. Explore the concept of rational exponents and understand how they are converted to radical expressions through several example problems.

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### Other Chapters

Other chapters within the Common Core Math - Number & Quantity: High School Standards course