Mathematics 9 Quarter 2 Module 7: Simplifying Radical Expressions Using the Laws of Radicals

Mathematics Quarter 2 – Module 7 Simplifying Radical Expressions Using the Laws of Radicals

9

Mathematics Quarter 2 – Module 7 Simplifying Radical Expressions Using the Laws of Radicals

[illustration: students reading and studying outdoors and at a table, with a globe and a lamp]

CO_Q2_Mathematics 9 _ Module 7

Mathematics – Grade 9 Alternative Delivery Mode Quarter 2 – Module 7: Simplyfying Radical Expressions Using the Laws of Radicals First Edition, 2020

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Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: Mimilanie F. dela Merced – TIII Mary Ann J. Espina – TII Jerrylyn I. Clemente – TI Queenie Pearl E. Domasig – Teacher II

Editors: Juvy G. Delos Santos – Head Teacher VI Sally C. Caleja – Head Teacher VI

Validators: Remylinda T. Soriano, EPS, Math Angelita Z. Modesto, PSDS George B. Borromeo, PSDS

Illustrator:

Layout Artist: Peter A. Alavanza

Management Team: Malcolm S. Garma, Genia V. Santos Dennis M. Mendoza, Maria Magdalena M. Lim Aida H. Rondilla, Lucky S. Carpio

Printed in the Philippines by _____

Department of Education – National Capital Region Office Address: Misamis St., Brgy. Bago Bantay, Quezon City Telefax: (632) 8926-2213 /8929-4330 /8920-1490 and 8929-4348 E-mail Address: ncr@deped.gov.ph

Mathematics Quarter 2 – Module 7 Simplifying Radical Expressions Using the Laws of Radicals

9

Mathematics Quarter 2 – Module 7 Simplifying Radical Expressions Using the Laws of Radicals

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

LEARNING COMPETENCY

The learners will be able to:

• Simplify radical expression using the laws of radicals
• Rationalize a fraction with radical in the denominator

What I Know

Find out how much you already know about the module. Choose the letterof the best answer. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.

1. When an exact answer is asked, we must express it as a radical expression in
   a. simplified form
   b. radical form
   c. exponential form
   d. rational form

   
   

2. Which of the following is a radical expression?
   a. 5
   b. -5
   c. x⁵
   d. √5

   
   

3. Numbers such as 4, 9, and 25 whose square roots are integers are called ______.
   a. perfect cube
   b. perfect square
   c. nth root
   d. roots

   
   

4. Numbers such as 8, 27, and 125 whose cube roots are integers are called _____
   a. perfect cube
   b. perfect square
   c. nth root
   d. roots

   
   

5. It is the process of eliminating the radical expression in the denominator.
   factorization
   a. factorization
   b. extracting the roots
   c. rationalization
   d. simplifying radicals

   
   

6. It is the reverse process of raising a number to the second power.
   a. rationalization
   b. simplifying
   c. extracting the roots
   d. squaring a number

   
   

7. Which of the following is a square root of 196?
   a. 14
   b. 24
   c. 96
   d. 98

   
   

8. Between what two consecutive whole numbers does √31 lie?
   a. 4 and 5
   b. 6 and 7
   c. 5 and 6
   d. 7 and 8

   
   

9. Find the square root of 64.
   a. 9
   b. 32
   c. 8
   d. 4096

   
   

10. The simplest form of ∛(40x⁸y⁹z¹⁰) is
    a. 2x²y³z³∛(5x²z)
    b. 3x²y³z³∛(5x²z)
    c. 2xyz∛(5xyz)
    d. 3xyz∛(5xyz)

    
    

11. Simplify √(121/49)
    a. 11/7
    b. 3/7
    c. 11/9
    d. 1/7

    
    

12. Simplify the expression √(28x³y²z / 7xy²)
    a. 4√(xz)
    b. 2x√z
    c. 2√(xz)
    d. 4x√z

    
    

13. Rationalize the denominator 4 / (3√(2x))
    a. 2√(2x) / (3x)
    b. 2 / √x
    c. 2 / x
    d. 2x

    
    

14. Simplify ∛(²√(x²⁴))
    a. √x
    b. x⁴
    c. x√x
    d. ¹²√x

    
    

15. What is the simplified form of ⁵√(32/x)?
    a. ⁵√x / x
    b. 2⁵√(x⁴) / x
    c. 2x
    d. X

    
    

Lesson 1 SIMPLIFYING EXPRESSIONS WITH RATIONAL EXPONENTS

You’ve learned about rational expressions and radical expressions from previous modules. This time, you will learn how to simplifying radical expressions using the laws of radicals.

What’s In

Am I PERFECT or NOT?

Simplify each radical and determine whether it has PERFECT nth root or NOT. Write P for perfect nth root and N if not.

1. √1
2. ±√49
3. −√81
4. ²√50 [?]
5. √242
6. ∛8
7. ∛(−64)
8. −∛108
9. ∛32
10. ∛(−1)

Do you still remember the Laws of Radicals?

If we have (⁵√x)⁶ this means, it is also equivalent to ⁵√x⁶.

How about √3 · √y? We may also write it as √3y. It has a somewhat same idea when we have √3 / √y then it can also be written as √(3/y).

While we also know that when the index and the exponent is the same, ∛z³ then we can simply write it as z.

This time, we will use the laws of radicals we learned in simplifying radicals.

What’s New

Communication and Critical Thinking)

INVESTIGATE 1

What are the ROOTS?

DIRECTION: Find the roots of each given expressions. Choose your answer from the basket. Use your answer to decode, what kind of fruits I have?

[illustration: tree with ten numbered fruits, each bearing a radical expression]

Expressions on the tree:

• Fruit 2: √(4a²b⁴)
• Fruit 3: −√(121x⁴y⁶z⁸)
• Fruit (unnumbered, left side): √(a⁴)
• Fruit (unnumbered, left side): 2r√(49s⁵t¹⁰)
• Fruit 5: −5√(250x⁷y¹¹)
• Fruit 6: ∛(125a³)
• Fruit 7: ∛(128s⁹t¹²)
• Fruit 8: ∛(−64d⁵e⁷f⁹)
• Fruit 9: −∛(−54d⁹ef)
• Fruit 10: −11ab∛(a¹²b¹⁸)

[transcriber’s note: the tree shows fruit numbers 1 to 10, but only some fruits carry a printed number next to their expression. The two expressions on the left side are placed beside fruits 1 and 4 and their numbers are not clearly printed, so they are listed without a fruit number.]

What is the name of my fruit?

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2 9 5 3 4 1 6 10 10 5

Basket choices:

Letter	Expression
S	a²
O	14rs²t⁵√s
G	−11x²y³z⁴
T	5a
A	3d³∛(2ef)
U	4s³t⁴∛2
E	−11a⁵b⁷
I	−4de²f³∛(d²e)
M	2ab²
N	−25x³y⁵√(10xy[?])

[transcriber’s note: the right edge of the N basket is cut off in the image.]

What is It

(Communication and Critical Thinking)

SIMPLIFYING RADICAL EXPRESSIONS

Let us study the examples below about simplifying radical expressions

Examples:

Simplify each expression.

1. √(a⁴) = a² since a^(4/2) = a² and a² • a² = a⁴

   
   

2. √(4a²b⁴c⁶) = (4a²b⁴c⁶)^(1/2)
   = 2^(2/2) a^(2/2) b^(4/2) c^(6/2)
   = 2ab²c³
   (note that 4 = 2²)

   
   

3. ∛(−54x⁴y⁸) = ∛((−27)(2)x³ • x • y⁶ • y²)
   = ∛((−3)³x³(y²)³) • ∛(2xy²) Factor out the perfect cube.
   = −3xy² ∛(2xy²) Simplify the perfect root.

   
   

Questions:

1. How did you answer the exercises?
2. Did you understand the process to simplify the given radical expressions?
3. Did you answer the exercises correctly?
4. What concepts or skills did you learn from the previous exercises?

Recall the Laws of Radicals

• ⁿ√(x^m) = (ⁿ√x)^m
• x^(m/n) = (ⁿ√x)^m = ⁿ√(x^m)
• ⁿ√x • ⁿ√y = ⁿ√(xy)
• ⁿ√x / ⁿ√y = ⁿ√(x/y)
• (ⁿ√x)^m = x

INVESTIGATE 2

Make me SIMPLE

DIRECTION: Write each radical in simplest form.

1. √(9/16)
2. ∛(8/27)
3. √25 / √100
4. −∛3 / ∛5
5. −√(4/5)
6. ∛(7/6)
7. 2 / √2
8. 3 / ∛2
9. −1 / (3√2)
10. −3 / (2∛3)

Questions:

1. How did you find the answers in the previous exercises?
2. Did you answer the exercises correctly?
3. Did you understand the process the get the correct the answer in each given?
4. What skills/concepts did you learn from the previous exercises?

SIMPLIFYING RADICAL EXPRESSIONS

Let us study the next set of examples about simplifying radical expressions.

Simplify.

1. √(75/36)

Solution: √(75/36) = √75 / √36 — By Quotient Law of Radicals = √25 √3 / 6 — By the Multiplication Law of Radicals = 5√3 / 6 — Since √25 = 5

2. √(18x³ / 49xy²)

Solution: √(18x³ / 49xy²) = √(18x² / 49y²) — Simplify = √18x² / √49y² — By Quotient Law of Radicals = √(9x² • 2) / √(49y²) — By the Multiplication property = 3x√2 / 7y — Since √(9x²) = 3x and √(49y²) = 7y

What if we are asked to simplify 1/√2 or √(1/2)? How do we simplify these?

Simplifying these kind of expressions is called the process of rationalizing the denominator.

In this case, to eliminate the radical in the denominator, we may multiply √2 on both numerator and denominator. Through that, the denominator will become a perfect square.

Multiplying √2 on both numerator and denominator is equivalent to multiplying the fraction by 1, therefore there will be no changes in the value.

DIRECTION:

From the box below choose the radicals that will help you rationalize the given expressions.

Box of choices:

∛(x²) ∛(a²) √(5x) ∛(9x) ²√(3b)
√m √(2ab²) ∛(5x²y²) ∛(x²y²) √(2b)

1. 1 / √(2b³)
2. x² / ∛x
3. ²√(5 / 3b)
4. 2a / ∛a
5. 2√5r / √(m³)
6. 8 / ∛(3x²)
7. √(32c⁵d³) / √(2ab)
8. 8 / ∛(3x²)
9. 6√(45y³) / 3√(5x)
10. ∛(5z⁷b⁴) / ∛(25xy)

TAKE NOTE: The choices given for this exercise are not the answers. These will be used to rationalize the given radical expression.

Questions:

1. How did you find the answers in the previous exercises?
2. Did you answer the exercises correctly?
3. Did you understand the process the get the correct the answer in each given?
4. What skills/concepts did you learn from the previous exercises?

A radical expression is not considered in simplest form when its denominator contains a radical. The process of rewriting the quotient so that the denominator doesn’t contain any radical expression is called rationalizing the denominator.

To rationalize the denominator, we multiply the denominator by an appropriate expression such that the product will be a perfect nth root.

Examples:

Rationalize the denominator.

Note: Multiply by √5/√5 since √5 • √5 = √25 = 5.

1. 1/√5 = (1/√5) • (√5/√5) = √5/√25 = √5 / 5
   Note: The quotient √5/5 has been rationalized.
   Note: Write the numerator and denominator as two separate square roots using the Quotient Rule for Radicals.

   
   

2. √(7/18) = √7 / √18
   = (√7 / √18) • (√18 / √18)
   = √126 / 18 = (√9 • √14) / 18
   = (3 • √14) / 18 = (3 • √14) / (3 • 6)
   = √14 / 6
   Note: To rationalize the denominator of a fraction containing a square root, simply multiply both the numerator and denominator by the denominator over itself.
   Note: Be sure to simplify the radical in the numerator completely by removing any factors that are perfect squares.
   Note: Be sure to also simplify the fraction by cancelling any common factors between the numerator and denominator.

   
   

3. 4 / (3√2x) = (4 / 3√2x) • (√2x / √2x)
   = 4√2x / (3√4x²) = 4√2x / (3(2x)) = 4√2x / 6x
   = 2√2x / 3x
   Note: Multiply by √2x / √2x since √2x • √2x = √(4x²) = 2x.
   Note: Multiply 2x and 3.
   Note: Remove common factors of 4 and 6.
   Note: The final answer should not contain any radicals in the denominator. Also, any radicals in the numerator should be simplified completely. And the fraction should be simplified as well.

   
   

4. 5y / ∜x⁴ = (5y / ∜x) • (∜x³ / ∜x³)
   = 5y∜x³ / ∜x⁴ = 5y∜x³ / x
   Note: Multiply by ∜x³ / ∜x³ since ∜x • ∜x³ = ∜x⁴ = x.
   Note: Simply both the numerator and denominator.

   
   

5. 3 / (√(y+2)) = (3 / √(y+2)) • (√(y−2) / √(y−2))
   = (3√y − 6) / (√(y²) − 4) = (3√y − 6) / (y − 4)
   Note: The denominator is a binomial, therefore we multiply it by its conjugate. The conjugate of a binomial is a binomial of the same terms but of different sign of its second term.
   Note: In this case, the conjugate of √y + 2 is √y − 2.
   Note: Multiply the numerators then denominators and simplify. Multiplying the conjugates would result to √(y²) + 0 + 0 − 4 or simply y − 4. Which gives us a binomial without the radical sign on both terms.

   
   

What’s More

A. Simplify each radical.

1. √(x⁵)
2. ∛(125x⁵y¹⁸)
3. ∛(b⁷)
4. ∛(56m¹⁰n¹²)
5. ²√(45x³y⁸)
6. ∛(−16x⁶y³z¹²)
7. √(60x⁴y⁵)
8. ⁴√(16x¹²y²¹)
9. −2x√(3xy³z⁷)
10. ⁴√(81p²⁵q³⁷)

B. Simplify.

1. 1 / √25
2. √(2 / 121)
3. √11 / (−√36)
4. √(1 / 49)
5. √(4 / 9)
6. −3 / (−√9)
7. √(5 / 4)
8. −√36 / √64
9. 3 / √49
10. −√(16 / 144)

C. Simplify by rationalizing the denominator.

1. √(2x / y)
2. √(5m / 7n²)
3. √(3m / 2n)
4. 1 / √(3ab)
5. √8x / √3y
6. 5 / √18y
7. 3√3 / √6m
8. √(2x / 5y)
9. 4√2 / √20p
10. √5a / √b

What I Have Learned

Simplifying Radicals

a. Removing Perfect nth Powers: ⁿ√(ab) = ⁿ√a • ⁿ√b

b. Reducing the index to the lowest possible order: ᵐ√(ⁿ√a) = ᵐⁿ√a = ⁿ√(ᵐ√a)

c. Rationalizing the denominator of the radicand

• The process of rewriting the quotient so that the denominator doesn’t contain any radical expression is called rationalizing the denominator.
• To rationalize the denominator, we multiply the denominator by an appropriate expression such that the product will be a perfect nth root.
• When the denominator is a binomial, we multiply it by its conjugate. The [illegible]

What I Can Do

DIRECTION: Draw a line to match Column A with the correct answer in Column B.

[transcriber’s note: the columns are printed side by side with no row alignment, so the rows below pair Column A item 1 with Column B item a and so on, for layout only. The answers are to be drawn by the learner.]

Column A	Column B
1. √(3³6³)	a. −2x¹⁸
2. ∛(2⁴5⁵)	b. 2⁵√(x⁴) / x
3. √(63m²)	c. 2√15 / 5
4. ∛(−40x⁹y¹⁰)	d. −2x³y³ ∛(5y)
5. −2x√(3xy³z⁷)	e. 10∛50
6. 6√5 / (5√3)	f. 54√2
7. 4 / ⁴√(8m²)	g. 3m√7
8. ⁵√(32 / x)	h. −2xyz³√(3xyz)
9. 5x⁴y² √((4/25)x²⁰y³⁰)	i. 2⁴√(2m²) / m
10. 20x⁶ ∛((−1/1000)x³⁶)[?]	j. 2x¹⁴y¹⁷

Assessment

Read each item carefully. Choose the letter of the correct answer.

1. Simplify √72. 2

   • a. 6√2
   • b. 2√6
   • c. 12

2. √98 is equal to

   • a. 7√2
   • b. 14
   • c. 7√8
   • d. 8

3. Find the root of ∛8

   • a. 3
   • b. 4
   • c. ±2
   • d. 2

4. Which of the following is equal to ∛(−729)

   • a. -9
   • b. 27
   • c. 81
   • d. ±9

5. ∛(1/8) when simplify is equal to

   • a. 1/4
   • b. 2
   • c. 1/∛2
   • d. 1/2

6. √(49x⁸) in simplified form

   • a. a√24.5
   • b. 7x⁸
   • c. 7x⁴
   • d. 7√(5x⁸)

7. Simplify √(250x⁴y⁵)

   • a. xy√125
   • b. 5√(10x⁴y⁵)
   • c. 5x²y²√10y
   • d. 5√(10x⁴y⁵)

8. Evaluate the expression √(160a⁵b⁴)

   • a. ab√80
   • b. 4√(10a⁵b⁴)
   • c. 4√(10a⁵b⁴)
   • d. 4a²b²√(10a)

CO_Q2_Mathematics 9 _ Module 7

9. Which of the following is equal with ∛√64 ?
   a. ⁶√64
   b. ³√64
   c. ²√64
   d. ³√8

   
   

10. It is the process of eliminating the radicals in the denominator of a fraction.
    a. Elimination
    b. Rationalization
    c. Fractionalization
    d. Simplification

    
    

11. Simplify √(a⁴b⁴/c³).
    a. √(a²b²/2)
    b. (a²b²/c²)√c
    c. (a²b²/c²)√c
    d. (a²b²/c²)√c²

    
    

12. Which of the following is equal to ⁸√(36a¹²b⁶)?
    a. ³√(6ab³)
    b. a⁴√(6ab³)
    c. a⁴√(6a²b³)

    
    

13. a⁴√(6ab⁴) Which of the following is true?
    a. ³√(3/x) = ³√(3x²)/x
    b. ³√(3/x) = ³√(x²)/x
    c. ³√(3/x) = ³√(3x)/x²
    d. ³√(3/x) = ²√(3x²)/x

    
    

14. Which of the following is equal to 3/⁵√2 ?
    a. 3⁵√16 / 2
    b. 3⁵√8 / 2
    c. 3⁵√2 / 16
    d. 3⁵√4 / 2

    
    

15. Which of the following needs to be rationalized?
    a. (a²b²/c²)√c
    b. a²b²√c / c
    c. (ab/c²)√c
    d. a²b² / √c

    
    

Additional Activities

Critical Thinking)

What you choose to do does not just take up your space, it also takes up your time and what takes up your time adds up to how you live your life. Simplify your lives. It increases beliefs and build confidence that may lead to your dreams and success!

Your thought about it:

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E-Search

To further explore the concept learned today and if it is possible to connect the internet, you may visit the following links:

• https://www.youtube.com/watch?v=4Gq3LPORQ-U (Simplifying Radical Expressions Adding, Subtracting, Multiplying, Dividing, & Rationalize)
• https://www.youtube.com/watch?v=BPY7gmT32XE (How to Simplify Radicals (Nancy Pi))
• https://www.youtube.com/watch?v=2YpE4_HIAn8 (Simplify a radical expression with variables)
• https://www.youtube.com/watch?v=X-ZAOc5gdFY (Simplifying Radical Expressions with variables)
• https://www.youtube.com/watch?v=Y1tDYBA5uIA (Rationalizing a Denominator)

PISA BASED WORKSHEET

MAKING MELODIES WITH MY TRIANGLE

To honor the front liners, Savannah is planning to serenade them using her favorite musical instrument. This musical instrument is a three-sided figure with equal sides. One of the sides measures √(18x³y) / √(2x⁴y²).

[illustration: a triangle musical instrument with a striker]

Let’s Analyze

1. What is Savannah’s favorite musical instrument?
2. What kind of polygon is her favorite musical instrument?
3. What do you mean by rationalizing the denominator?
4. How will you know if the given radical is in simplified form?
5. Find the simplified form of √(18x³y) / √(2x⁴y²)?

Answer Key

[illustration: answer key emblem]

transcriber’s note: this page is printed upside down (rotated 180 degrees). The entries below were read from the rotated image; entries marked [?] are uncertain.

WHAT I KNOW

1. A
2. D
3. B
4. A
5. C
6. C
7. A
8. C
9. C
10. A
11. A
12. B
13. A
14. B
15. B [?]

WHAT’S IN

1. I Perfect
2. 4√7 Perfect [?]
3. -9 Perfect
4. 5√2 Not
5. 11√7 Not [?]
6. 2 Perfect
7. -4 Perfect
8. -3∛4 Not [?]
9. 2∛4 Not [?]
10. -1 Perfect

WHAT’S NEW

Investigate 1: WHAT’S MY PROOFS

transcriber’s note: the letter-and-number grid for this activity is rotated and partly unclear; the answer choices are not reliably readable, so only the expressions that could be read are given.

1. a²
2. 2ab²
3. 4√5
4. 2 (a) [?]
5. 5a
   1. [?]
6. 4s³t⁴√2 [?]
7. 3√(4t)
8. −4de²√(5a²)? [?]
9. −25x³y³√(10xy)? [?]

Investigate 2: MAKE ME SIMPLE

1. ¾
2. 1/ 4? [?]
3. −2√6 / 5 [?]
4. √2
5. −√2 / 5 [?]
6. 2/3 [?]
7. −∛75 / 5 [?]
8. ⁴√(25/2)?
9. 3∜4 / 2 [?]
10. ∛9 / 2 [?]

Investigate 3: The CLUE

1. ∛2 / b? [?]
2. 2b² / (∛b) [?]
3. 4√5 [?]
4. 4a²·³√b / b [?]
5. 5x [?]
6. x²y²/(∛x²)? [?]
7. ∛x… [?]
8. ∛(3x… ) [?]
9. 3y²? [?]
10. (x²y²·∛(xy))/25? [?]

WHAT’S MORE A

1. x²√x
2. b²∛b / … [?]
3. 3xy⁴ / √(5x) [?]
4. 2x²y²√(15y) [?]
5. −7y²z²√(3xyz) [?]
6. 5xy⁵·∜x² [?]
7. 2m²n²√(7m) [?]
8. −2x³y²z²√(2) [?]
9. 2x³y⁵√y [?]
10. 3p⁶q⁴√(pq) [?]

WHAT’S MORE B

1. 1/5
2. √6 / … [?]
3. √6 [?]
4. 3 / … [?]
5. 11/√3 [?]
6. 2/3
7. −3/4
8. −1/3
9. −6/√11
10. 1

WHAT’S MORE C

transcriber’s note: the expressions in this section are rotated and mostly unclear; they are not reproduced.

[illegible]

WHAT I CAN DO

1. F
2. E
3. G
4. D
5. H
6. C
7. I
8. B
9. J
10. A

ASSESSMENT

1. B
2. A
3. D
4. A
5. D
6. C
7. C
8. D
9. A
10. B
11. B
12. C
13. A
14. A
15. D

PISA BASED

1. Savannah’s favorite musical instrument is triangle instrument.
2. It is a polygon classified as a triangle.
3. Rationalizing the denominator is the process of rewriting the quotient so that the denominator contains no nth roots.
4. • The radicand has no factors except 1
   • No fraction appears in a radicand
   • No radical appears in the denominator of a fraction
5. ∛(xy) / x [?]

References

Gladys C. Nievera and Minie Rose C. Lapinid, 2013. Grade 9 Mathematics Patterns and Practicalities K to 12 Edition. Don Bosco Press, Makati City/Philippines

Orlando A. Oronce and Marilyn O. Mendoza, 2014. E-math 9 Seamless K-12 Edition. Rex Book Store, Manila/Philippines

DepEd-IMCS, 2014. Mathematics Learners Material 9 First Edition. Department of Education, Pasig City/Philippines

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph