Simplifying Radicals

Algebra Simplifying Radicals StAIR Project Lori Ferrington

Students will know the properties of positive and negative roots. (A1.1.2) • Students will know how to simplify positive and negative radicals for later use in algebraic equations. (A1.1.2) • Students will know how to multiply radical expressions for later use in algebraic equations. (A1.1.2) • Michigan Department of Education - High School Content Expectations Objectives and Standards Simplifying Radicals

This lesson is designed for Mrs. Ferrington’s Algebra class. • You are to navigate your way through this lesson individually. • The home page will allow you to navigate through different parts of this lesson • If you link out to a webpage, close the window when you are done viewing the content on the webpage to return to this lesson. • Key terms in this lesson will be shown in red. • After this lesson, you will be given an assessment to measure your understanding. • Have Fun! :) • By clicking on this icon, you can access the definitions of key terms at any time during this lesson. Introduction Click to return to the home page Click to go on to the next page Click to return to the previous page Simplifying Radicals

A prime number is only divisible by 1 and itself. • A factor is a number that can divide another number without a remainder. • 2 and 3 are prime factors of 6 because 23=6 and both 2 and 3 are prime. 180 10 18 5 2 3 6 Warm up 2 3 So, we can say that the prime factorization of 180 is 22335. Let’s review factor trees and prime factorization.

Video explanation here • Extra practice here • You think you’re ready…click the next button to continue. Warm Up Need extra practice with factor trees?

Which of the following are factors of 252? A.) 2126 252 B.) 2118 C.) 386 D.) 551 Warm UP Question #1 Complete the factor tree below.

Great job!2 and 126 are factors of 252. • Which of the following are factors of 126? 252 2 126 A.) 264 B.) 267 C.) 914 D.) 719 WARM up Question #1 Complete the factor tree below.

2 and 126 are factors of 252. • Fabulous!9 and 14 are factors of 126. • Which of the following are factors of 9? 252 2 126 A.) 23 9 14 B.) 33 Warm Up Question #1 C.) 22 D.) 34 Complete the factor tree below.

2 and 126 are factors of 252. • 9 and 14 are factors of 126. • Excellent! 3 and 3 are factors of 9. • Which of the following are factors of 14? 252 2 126 9 14 A.) 23 Warm Up Question #1 3 3 B.) 25 C.) 37 D.) 27 Complete the factor tree below.

2 and 126 are factors of 252. • 9 and 14 are factors of 126. • 3 and 3 are factors of 9. • Awesome! 2 and 7 are factors of 14. 252 2 126 9 14 Warm Up Question #1 3 2 3 7 The prime factorization of 252 is 22337. Great job! Click the next arrow to try another. Complete the factor tree below.

Some common prime numbers are 2, 3, 5, 7, 11, and 13. A.) 2357 112 B.) 22227 C.) 2247 D.) 23335 Warm up question #2 Find the prime factorization of 112.

Great Job! You’re right!! 112 22227 = 112 2 56 8 7 Warm up question #2 2 2 4 2 Now let’s move on to some new stuff. Find the prime factorization of 112.

Home • A radical is any expression that contains a square root, cube root, etc. • The symbol representation of a radical is √ . • The term radical comes from the late Latin radicallis meaning “of roots” and from Latin radix meaning “root.” We say that a radical expression is simplified, or in its simplest form, when the radicand has no square factors. There are three parts to the following lesson that will teach you about simplifying radicals. Please follow them in order. You will be able to navigate back to review any lessons again. I. Positive Radicals Simplifying Radicals II. Negative Radicals III. Product of Two Radicals Quiz Home

Check for a perfect square • Find the prime factorization of the radicand • Identify pairs of primes in radicand • Simplify perfect squares √12 = 3.4641; this is not a perfect square. √12 = √223 = √22  √3 = √4  √3 = 2√3 12 2 6 2 3 i. Positive radicals So √12 simplifies to 2√3. Example #1: Simplify √12

√162 = 12.7279; this is not a perfect square. 162 √162 = √23333 = √2  √33  √33 = √2  √9  √9 = 33 √2 = 9√2 2 81 9 9 • Be sure that any perfect squares come out in front (to the left) of the radical symbol. This prevents confusing it with numbers in the radicand. I. positive radicals 3 3 3 3 Example #2: Simplify √162

Simplify 2√605 A.) 11√5 • Hint: If an integer is in front of the radical, do not move it. B.) 11√2 C.) 10√11 D.) 22√5 i. Positive radicals For extra examples go here Check for Understanding…

Try Again! Oops, 2√605 ≠ 11√5 Don’t forget the integer in front of the radical symbol i. Positive radicals Check for Understanding…

Try Again! Sorry, 2√605 ≠ 11√2 You can only square root perfect squares. Look for pairs of numbers in the radicand. i. Positive radicals Check for Understanding…

Try Again! Bummer, 2√605 ≠ 10√11 You can only square root perfect squares. Look for pairs of numbers in the radicand. i. Positive radicals Check for Understanding…

Great Job! Yes!! 2√605 = 2√51111 = 2 √5  √1111 = 2 √5  √121 = 211 √5 = 22√5 i. Positive radicals Now on to negative radicals in section II … Check for Understanding…

Let’s look at example #1: Simplify -2√72 Which do you think is the correct answer? • Hint: many of the same properties of simplifying positive radicals apply in this situation. A.) 4√6 B.) -12√2 C.) 12√3 ii. negative radicals D.) -8√3 What do you think happens if a negative is outside of the radical?

Example #1: Simplify -2√72 You’re correct! The answer is B.) -12√2, but why? What rule best fits when simplifying radicals with a negative in front of them? A.) A negative in front of the radical goes in the radicand. II. negative radicals B.) A negative in front of the radical stays in front. C.) A negative in front of the radical cannot be simplified. What do you think happens if a negative is outside of the radical?

√50 = 7.017; this is not a perfect square. • Don’t forget the rule you determined in the previous slide: • A negative in front of the radical stays in front. 50 -5√50 = -5√255 = -5  √2  √55 = -5  √2  √25 = -55 √2 = -25√2 2 25 5 5 II. Negative radicals Example #2: Simplify -5√50

Simplify -2√98 A.) -14√2 B.) -7√2 C.) -2√7 D.) -4√7 II. Negative Radicals For extra examples go here Check for understanding…

Try Again! Oh no, -2√98 ≠ -7√2 Don’t forget about the integer in front of the radical iI. Negative radicals Check for Understanding…

Try Again! Sorry, -2√98 ≠ -2√7 Look for pairs of numbers in the radicand. iI. Negative radicals Check for Understanding…

Try Again! Nice try, -2√98 ≠ -4√7 Look for perfect squares in the radicand. iI. Negative radicals Check for Understanding…

Fantastic! You got it!! -2√98 = -2  √277 = -2 √2  √77 = -2 √2  √49 = -27 √2 = -14√2 iI. Negative radicals Let’s look at the multiplying two radicals in section III … Check for Understanding…

Check for perfect squares. • Multiply the numbers in front of the radical and multiply the radicands. • Simplify the radicand by finding the factorization or by identifying perfect squares Neither √3 nor √12 are perfect squares. 2√3  √12 = 2 √312 = 2 √36 = 26 = 12 III. Product of 2 radicals So 2√3  √12 simplifies to 12. Example #1: Simplify 2√3  √12

Neither √3 nor √8 are perfect squares. 3√32√8 = 32 √38 = 6 √24 = 6 √2223 = 6 √22  √23 = 6  √4 √6 = 6  2 √6 = 12√6 • Remember to use factor trees to help you find the prime factorization of the radicand. III. Product of 2 radicals Example #2: Simplify 3√3  2√8

Simplify √10  √5 A.) 5√5 B.) 2√5 C.) -2√5 D.) 5√2 III. Product of 2 radicals For extra examples go here Check for understanding…

Try Again! Oh no, √10  √5 ≠ 5√5 Look for perfect squares in the radicand. III. Product of 2 radicals Check for Understanding…

Try Again! Not that one, √10  √5 ≠ 2√5 Look for perfect squares or pairs of integers in the radicand. III. Product of 2 radicals Check for Understanding…

Try Again! Whoops, √10  √5 ≠ -2√5 Be careful of your use of negatives. III. Product of 2 radicals Check for Understanding…

Awesome! That’s correct! √10  √5 = √105 = √50 = √255 = √2  √55 = √2  √25 = 5√2 III. Product of 2 radicals Click next to take the Simplifying Radicals Quiz … Check for Understanding…

QUIZ • Move through this quiz by selecting the correct simplified form of the radical expression given. • You must get each problem correct before proceeding to the next question. • Good luck! Quiz Click here to begin the quiz • Don’t forget your factor trees!  Instructions

1 A.) 20√2 B.) 10√5 C.) 10√2 D.) 2√5 QUIZ 1. Simplify √200

Hint: Find your prime factorization of 200, and then look for perfect squares. QUIZ Oops, go back and try again! 1. Simplify √200

2 A.) 10√10 B.) 100√10 C.) 10√100 D.) 20√10 Quiz 2. Simplify 10√1000

Hint: Don’t forget the 10 in front of the radical. Quiz Oops, go back and try again! 2. Simplify 10√1000

3 A.) -4√8 B.) 4√4 C.) -8√2 D.) -4√2 Quiz 3. Simplify -√32

Hint: Remember the rule about negatives in front of the radical symbol. Quiz Oops, go back and try again! 3. Simplify -√32

4 A.) -108 B.) -36√3 C.) -81 D.) -√93 Quiz 4. Simplify -3√27  4√3

Hint: Multiply your integers in front of the radical and the multiply the radicands before you simplify. Quiz Oops, go back and try again! 4. Simplify -3√27  4√3

Great! You did it! Fabulous! Superb Excellent

Prime number • Factor • Prime Factorization • Radical • Radicand • Perfect Square A number only divisible by only 1 and itself A number that can divide another number without a remainder. The list of all of the prime numbers whose product makes up a number. Any expression that contains a root (e.g. square root). The symbol is √ The number underneath the root or radical symbol. It is the product of some integer with itself (e.g. √9 = 3). Vocabulary Terms and definitions (in order of appearance)

Simplifying Radicals