# MA004: Intermediate Algebra

Unit 2: Simplifying Radical Expressions   Now that you understand linear equations, the next step is to raise the exponent of the variable from 1 to a fraction. This is called a radical. This unit will introduce you to radicals and the equivalent form of radicals – rational exponents. You will learn the arithmetic of radicals, including rationalizing denominators that contain radicals. There are many important applications of radicals, such as the Pythagorean Theorem in geometry, which uses square roots. In addition, financial calculations of interest rates and current values rely on rational exponents.

Completing this unit should take approximately 22.75 hours.

☐    Subunit 2.1: 4.25 hours

☐    Subunit 2.2: 2 hours

☐    Subunit 2.3: 4.25 hours

☐    Subunit 2.4: 4.25 hours

☐    Subunit 2.5: 4 hours

☐    Subunit 2.6: 4 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - simplify expressions with rational exponents; - simplify radical expressions; - rationalize denominators (monomials and binomials) of radical expressions; - add, subtract, and multiply radical expressions with and without variables; - reduce the index on a radical; and - combine radicals of mixed index.

2.1 Simplifying Radicals   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.1: Square Roots” and “Chapter 8, Section 8.2: Higher Roots” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.1: Square Roots” (PDF) and “Chapter 8, Section 8.2: Higher Roots” (PDF)

Instructions: Read Sections 8.1 and 8.2 in Chapter 8 of your textbook, pages 288–294, to learn about square roots and higher roots. The square root is probably the second most common power (after the linear equations). If you took Geometry in high school, you are familiar with the square root from the Pythagorean Theorem. Note that this reading also covers all the topics in Subunits 2.1.1–2.1.5.

Reading this section and taking notes should take approximately 2 hours.

• Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Simplify Radicals” Link: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Simplify Radicals” (PDF)

Instructions: Complete pages 28–32 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 2.1.1–2.1.5, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 1 hour.

2.1.1 Prime Factorization   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.1.

Instructions: Watch the video linked above, which discusses prime factorization. Prime factorization breaks a positive integer down into a product of prime factors, such as 168 = 23∙3∙7. The purpose of prime factorization is to remove even powers of a number from under the radical sign.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.1.2 Dividing the Exponent by the Index (Perfect Roots)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.1.

Instructions: Watch the video linked above, which discusses radicals. If the prime factorization of the number has all even powers, then this video will show how to remove the radical sign.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.1.3 Dividing the Exponent by the Index (Not Perfect Roots)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.1.

Instructions: Watch the video linked above, which discusses simplifying radicals that are not a perfect root. You have a square of a number that is not a perfect square but has a perfect square as a factor. You want to have the number under the square root as small as possible.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.1.4 Simplifying Radicals with Coefficients   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.1.

Instructions: Watch the video linked above, which discusses what you do with the coefficient to a radical after simplifying the radical. Remember to multiply the coefficient by any number removed from the radical.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.1.5 Simplifying Radicals with Variables   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.1.

Instructions: Watch the video linked above, which discusses radicals with variables. Basically, each variable will act the same as a prime factor.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.2 Add/Subtract Radicals   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.3: Adding Radicals” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.3: Adding Radicals” (PDF)

Reading this section and taking notes should take approximately 1 hour.

Instructions: Complete pages 33–34 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 2.2.1–2.2.2, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 30 minutes.

2.2.1 Combining Radicals   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.2.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.2.2 Simplifying and Combining Radicals   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.2.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.3 Multiplying Radicals   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.4: Multiply and Divide Radicals” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.4: Multiply and Divide Radicals” (PDF)

Instructions: Read Section 8.4, pages 298–302, in Chapter 8 of your textbook to learn how to multiply and divide two radicals. As with arithmetic, after learning how to add and subtract radicals, you want to multiply and divide radicals. These operations are substantially more complicated than with integers. Note that this reading also covers all the topics in Subunits 2.3.1–2.3.5.

Reading this section and taking notes should take approximately 2 hours.

• Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Multiplying Radicals” Link: Tyler Wallace’s Intermediate Algebra Lab Notebook“Multiplying Radicals” (PDF)

Instructions: Complete pages 35–39 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunit 2.3.1–2.3.5, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 1 hour.

2.3.1 Multiplying Monomials   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.3.

Instructions: Watch the video linked above, which discusses multiplying monomial radicals – multiplying two expressions that have a single radical in each. Remember to reduce the radical after multiplication of the radicals.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.3.2 Multiplying a Binomial by a Monomial   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.3.

Instructions: Watch the video linked above, which discusses the distributive rule with radicals. Recall the distributive rule of multiplication over addition from arithmetic:

a(b + c) = (a∙b) + (a∙c)

This video reminds us of this rule and applies it to when a, b, and c are radicals.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.3.3 Multiplying Two Binomials   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.3.

Instructions: Watch the video linked above, which discusses multiplying two binomial expressions using FOIL with radicals. Recall that when multiplying two binomial expressions, four terms result. To remember which products result in each term, FOIL (first, outer, inner, last) is a convenient acronym.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.3.4 Conjugates of Radicals   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.3.

Instructions: Watch the video linked above, which discusses conjugates of radicals. When dividing a radical expression by another radical expression, you want to end up with an expression that does not have a radical expression in the denominator. To accomplish this you must multiply the numerator and denominator by the conjugate of the radical expression in the denominator.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.3.5 Squaring a Binomial with Radicals   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.3.

Instructions: Watch the video linked above, which discusses squaring a binomial with radicals. Recall that

(a + b)2 = a2 + 2ab + b2

This video reminds us of this equation and shows how it works when a and b are radicals.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.4 Rationalizing Radicals   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.5: Rationalize Denominators” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.5: Rationalize Denominators” (PDF)

Instructions: Read Section 8.5 in Chapter 8 of your textbook, pages 303–309, to learn how to rationalize denominators. In this section, you will learn how to simplify radical expressions with a denominator. You first need to simplify all radicals. If no radicals are in the denominator, then you can use rules similar to arithmetic rules to simplify. If the denominator does have a radical expression, then you want to multiply the numerator and the denominator by the conjugate of the denominator and then simplify. This material also covers Subunits 2.4.1–2.4.5.

Reading this section and taking notes should take approximately 2 hours.

• Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Rationalize Denominators” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook“Rationalize Denominators” (PDF)

Instructions: Complete pages 40–44 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 2.4.1–2.4.5, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 1 hour.

2.4.1 Simplifying Radicals and Fractions   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.4.

Instructions: Watch the video linked above, which discusses simplifying the radical before reducing the fraction. Sometimes, simplifying the radical will result in a common factor in the numerator and denominator.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.4.2 Quotient Rule   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.4.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Quotient Rule” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Quotient Rule” (YouTube)

Instructions: Watch the video linked above, which discusses reducing a radical with a fraction. Whenever there is a radical divided by another radical, it is preferred to represent the fraction without a radical in the denominator.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.4.3 Monomial Square Roots   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.4.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Monomial Square Roots” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Monomial Square Roots” (YouTube)

Instructions: Watch the video linked above, which discusses rationalizing a fraction when there is a single radical in the denominator. This video gives additional examples similar to the previous video.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.4.4 Monomial Higher Roots   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.4.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Monomial Higher Roots” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Monomial Higher Roots” (YouTube)

Instructions: Watch the video linked above, which discusses rationalizing a higher root in a fraction. This extends the previous lesson by having a higher root than the square root in the denominator.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.4.5 Rationalizing Denominators with Binomials in the Denominator   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.4.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Binomial Denominators” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rationalize Denominators – Binomial Denominators” (YouTube)

Instructions: Watch the video linked above, which discusses rationalizing a fraction when the denominator is a binomial with radicals. Here the denominator must be rationalized by multiplying the numerator and the denominator by the conjugate of the denominator.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.5 Rational Exponents   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.6: Rational Exponents” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.6: Rational Exponents” (PDF)

Instructions: Read Section 8.6 in Chapter 8 of your textbook, pages 310–313, to learn the definition of rational exponents and the arithmetic of these exponents. Basically, rational exponents are equivalent to radicals with the denominator of the exponent being the index of the radical. For instance, √x is equivalent to x1/2. This material also covers Subunits 2.5.1–2.5.4.

Reading this section and taking notes should take approximately 2 hours.

• Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Rational Exponents” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook“Rational Exponents” (PDF)

Instructions: Complete pages 45–47 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 2.5.1–2.5.4, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 1 hour.

2.5.1 Converting Rational Exponents   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.5.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents - Convert” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponent - Convert” (YouTube)

Instructions: Watch the video linked above, which discusses how to convert from a radical to a rational exponent. A radical is equivalent to a rational exponent where the index of the radical is the denominator of the rational exponent.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.5.2 Evaluating Rational Exponents   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.5.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Evaluate” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Evaluate” (YouTube)

Instructions: Watch the video linked above, which discusses how to evaluate – determine the value of – radical expressions. You will see that rational exponents are easier to evaluate than radicals because the calculator can be used more readily.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.5.3 Simplifying Rational Exponents (Part 1)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.5.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Simplify (Part 1)” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Simplify (Part 1)” (YouTube)

Instructions: Watch the video linked above, which discusses simplifying expressions with rational exponents. This video first recalls the laws of exponents and then applies these laws to rational expressions with rational exponents.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.5.4 Simplifying Rational Exponents (Part 2)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.5.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Simplify (Part 2)” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational Exponents – Simplify (Part 2)” (YouTube)

Instructions: Watch the video linked above, which discusses simplifying expressions with rational exponents. This video gives a slightly more complicated example of applying the law of exponents to rational expression with rational exponents.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.6 Radicals of Mixed Index   - Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.7: Radicals of Mixed Index” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 8, Section 8.7: Radicals of Mixed Index” (PDF)

Instructions: Read Section 8.7 in Chapter 8 of your textbook, pages 314–317, to learn how to handle expressions containing mixed indices. As an example, to multiply a square and a cube root, both must be changed to a common index first, in this case, the sixth root. This material also covers Subunits 2.6.1–2.6.4.

Reading this section and taking notes should take approximately 2 hours.

• Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Mixed Index” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook“Mixed Index” (PDF)

Instructions: Complete pages 48–50 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 2.6.1–2.6.4, and then review the worksheet as you follow along with the videos for solutions.

Completing this assessment should take approximately 1 hour.

2.6.1 Reducing the Index   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.6.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Reduce Index” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Reduce Index” (YouTube)

Instructions: Watch the video linked above, which discusses reducing the index. Before simplifying expressions with mixed indices, you need to be sure that the index is as small as possible.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.6.2 Multiplying Radicals with Mixed Indices (Part 1)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.6.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Multiply (Part 1)” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Multiply (Part 1)” (YouTube)

Instructions: Watch the video linked above, which discusses multiplying expressions with mixed indices. Find a common index and convert both expressions to this common index.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.

2.6.3 Multiplying Radicals with Mixed Indices (Part 2)   Note: This subunit is also covered by the reading and assessment assigned in Subunit 2.6.

• Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Multiply (Part 2)” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Mixed Index – Multiply (Part 2)” (YouTube)

Instructions: Watch the video linked above, which presents another example of multiplying expressions with mixed indices.

You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take approximately 15 minutes.