Radical Operations: Simplify, Add, Subtract, Multiply

Algebra 2 Grades High School 4:12 Video
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Lesson Description

Master performing operations on radicals, including simplifying, adding, subtracting, and multiplying radical expressions. Learn to identify and combine like radicals and tackle more complex problems.

Video Resource

Performing Operations on Radicals

Mario's Math Tutoring

Duration: 4:12
Watch on YouTube

Key Concepts

  • Simplifying radicals
  • Combining like radicals
  • Distributive property with radicals
  • FOIL method with radicals

Learning Objectives

  • Students will be able to simplify radical expressions by factoring out perfect squares.
  • Students will be able to add and subtract radical expressions by combining like radicals.
  • Students will be able to multiply radical expressions using the distributive property and FOIL method.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a radical and the concept of perfect squares. Briefly discuss why simplifying radicals is important. Show the video 'Performing Operations on Radicals' by Mario's Math Tutoring. Ask students to take notes on the examples provided.
  • Simplifying Radicals (15 mins)
    Work through examples of simplifying radicals. Emphasize the process of finding the largest perfect square factor of the radicand. For example, simplify √18 as √(9*2) = √9 * √2 = 3√2. Provide several practice problems for students to work on individually or in pairs.
  • Adding and Subtracting Radicals (15 mins)
    Explain that radicals can only be added or subtracted if they are 'like radicals' (same radicand). Work through examples like 5√3 - 2√3 = 3√3. Include examples where students must first simplify the radicals before combining them, such as √18 + √20 - √8 = 3√2 + 2√5 - 2√2 = √2 + 2√5. Provide practice problems.
  • Multiplying Radicals (15 mins)
    Demonstrate how to multiply radicals using the distributive property and the FOIL method. Example: 2√3 * (4 - 5√3) = 8√3 - 10√9 = 8√3 - 30. Also, show how to multiply binomials containing radicals: (3 - 2√5) * (4 + 5√7) = 12 + 15√7 - 8√5 - 10√35. Provide increasingly challenging practice problems.
  • Wrap-up and Assessment (10 mins)
    Review the key concepts covered in the lesson. Administer the multiple-choice and fill-in-the-blank quizzes to assess student understanding. Answer any remaining questions.

Interactive Exercises

  • Radical Relay Race
    Divide the class into teams. Each team receives a set of radical expressions to simplify, add/subtract, or multiply. The first team to correctly solve all the problems wins.
  • Online Radical Calculator
    Have students use an online radical calculator (after they have attempted the problems themselves) to check their work and explore more complex radical expressions.

Discussion Questions

  • Why is it important to simplify radicals before performing other operations?
  • How are adding and subtracting radicals similar to combining like terms in algebraic expressions?

Skills Developed

  • Simplifying expressions
  • Applying the distributive property
  • Problem-solving

Multiple Choice Questions

Question 1:

Which of the following is the simplified form of √32?

Correct Answer: 4√2

Question 2:

What is the result of 3√5 + 7√5?

Correct Answer: 10√5

Question 3:

Simplify: √27 - √12

Correct Answer: √3

Question 4:

What is √3 * √12?

Correct Answer: √36

Question 5:

Simplify: 2√3 * (√3 + 1)

Correct Answer: 6 + 2√3

Question 6:

Which of the following are 'like radicals'?

Correct Answer: 2√5 and 3√5

Question 7:

What is the value of (√5)²?

Correct Answer: 5

Question 8:

Simplify: (2 + √3)(2 - √3)

Correct Answer: 4 - √9

Question 9:

Simplify √75 + √48 - √27

Correct Answer: 12√3

Question 10:

Which of the following is the correct expansion of (√2 + √3)²?

Correct Answer: 5 + 2√6

Fill in the Blank Questions

Question 1:

To simplify a radical, look for the largest ______ ______ that divides evenly into the radicand.

Correct Answer: perfect square

Question 2:

Radicals can only be added or subtracted if they have the same ______.

Correct Answer: radicand

Question 3:

When multiplying radicals, multiply the numbers outside the radical with each other and the numbers ______ the radical with each other.

Correct Answer: inside

Question 4:

√45 simplifies to ______.

Correct Answer: 3√5

Question 5:

5√2 + 3√2 - √2 = ______.

Correct Answer: 7√2

Question 6:

√8 * √2 = ______.

Correct Answer: 4

Question 7:

2√5(√5 - 3) = ______.

Correct Answer: 10 - 6√5

Question 8:

The product of (√7 + 2)(√7 - 2) is ______.

Correct Answer: 3

Question 9:

Like radicals have the same index and ______.

Correct Answer: radicand

Question 10:

When squaring a binomial containing a radical, remember to ______ the binomial.

Correct Answer: FOIL

Educational Standards

A2.N.1.3:Understand and apply the relationship between rational exponents to integer exponents and radicals to solve problems. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions. A2.A.2.5:Rewrite algebraic expressions involving radicals and rational exponents using the properties of exponents.

Teaching Materials

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