2. Introductory Message
For the facilitator:
This module was collaboratively designed, developed and evaluated by the Development and
Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the
standards set by the K to 12 Curriculum while overcoming their personal, social, and economic
constraints in schooling.
As a facilitator, you are expected to orient the learners on how to use this module. You also
need to keep track of the learners' progress while allowing them to manage their own learning.
Furthermore, you are expected to encourage and assist the learners as they do the tasks
included in the module.
For the learner:
This module was designed to provide you with fun and meaningful opportunities for guided
and independent learning at your own pace and time. You will be helped to process the
contents of the learning resource while being an active learner.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the module.
Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let’s Try before moving on to the other activities included in the
module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and in checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not hesitate to
consult your teacher or facilitator. Always bear in mind that you are not alone.
We hope that through this material, you will experience meaningful learning and gain deep
understanding of the relevant competencies. You can do it!
3. Let’s Learn
This module was designed and written with you in mind. It is here to help you master Laws of
Radicals. The scope of this module permits it to be used in many different learning situations.
The language used recognizes the diverse vocabulary level of students. The lessons are
arranged to follow the standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using.
After going through this module, you are expected to
1. derive the product law of radicals and;
2. simplify radical expressions using the product law of radicals.
Let’s Try
Directions: Read each question carefully and solve if necessary. Choose the letter of the
correct answer and write it before the number.
1. What is the simplified form of √12?
A. 2 B. 2√3 C. 6 D. 23
2. What is the greatest perfect cube factor of 81?
A. 1 B. 9 C. 27 D. 81
3. Simplify �48𝑥𝑥5𝑦𝑦2
A. 4𝑥𝑥3
𝑦𝑦√3 B. 4𝑥𝑥2
𝑦𝑦√3 C. 4𝑥𝑥𝑥𝑥√3𝑥𝑥 D.4𝑥𝑥2
𝑦𝑦√3𝑥𝑥
4.The following expressions can be simplified using Product Law of Radicals EXCEPT
A. √12 B. √5 ∙
3
√5 C. √2
3
∙ √12
3
D. 18
1
2
5. Which of the following equation is true?
A. √32 = 16√2 B.√𝑥𝑥8 = 𝑥𝑥6
C.√7 ∙ √5 = 35 D. √5𝑎𝑎 ∙ √15𝑏𝑏 = 5√3𝑎𝑎𝑎𝑎
6. What is the greatest perfect square factor of 18?
A. 2 B. 3 C. 9 D. 18
7. Simplify �√81
A. 3 B. 6 C. 9 D. 81
4. 8. Rewrite ���𝑦𝑦
3
as single radical.
A. �𝑦𝑦
12
B. �𝑦𝑦
8
C.�𝑦𝑦
5
D. �𝑦𝑦
9. Simplify √8𝑎𝑎5
3
A. 4√𝑎𝑎
3
B. 2𝑎𝑎4
√2𝑎𝑎
3
C. 2𝑎𝑎√𝑎𝑎2
3
D. 4𝑎𝑎
10. Simplify �√2
3
�
9
A. √2
3
B. 8 C. √2
27
D. 4
https://forms.gle/oCiJPt99LoF8oX7g8
5. Lesson
5 Product Law of Radicals
Let’s Recall
A. Evaluate: B. Complete the table
1. √25
Number
Factors
Greatest Perfect
Square/Cube
Another Factor
Ex 12 𝟐𝟐𝟐𝟐 3
1. 18 ____2
2. 50 ____2
3. 45 ____2
4. 54 ____3
5. 250 ____3
2. √9
3. √121
4. √27
3
5. √16
4
Since 49 𝑚𝑚 is equal to 0.049 𝑘𝑘𝑘𝑘, we write
𝑑𝑑 = 112.88√ℎ
Let’s Explore
Vladimir and Rodrigo flew using a hot air
balloon above the majestic Taal Volcano. The
view from above was as magnificent as they
thought it would be. At a height of 49 meters
above the ground, how far to the horizon were
they?
The approximate distance to the horizon (𝑑𝑑) in
kilometers is given by the equation
𝑑𝑑 = 112.88√ℎ
where ℎ is the altitude of the object in
kilometers.
6. 𝑑𝑑 = 112.88√0.049
𝑑𝑑 ≈ 24.99 𝑘𝑘𝑘𝑘
Thus, to the nearest hundredths, their distance to the horizon was 24.99 𝑘𝑘𝑘𝑘. If the situation
calls for an exact answer, we must express it as a radical expression in simplified form. To
simplify radicals, we will use the multiplication and division law of radicals.
Let’s Explain, Analyze & Solve
The Product Law of Radicals
If a and b represent a nonnegative real numbers and n is a positive integer,
√𝒂𝒂𝒂𝒂
𝒏𝒏
= √𝒂𝒂
𝒏𝒏
∙ √𝒃𝒃
𝒏𝒏
Thus, the nth root of the product of two nonnegative number is equal to the product of their
nth root.
Proof:
√𝑎𝑎𝑎𝑎
𝑛𝑛
= (𝑎𝑎𝑎𝑎)
1
𝑛𝑛 Express 𝒂𝒂𝒂𝒂 raised to rational exponent
= 𝑎𝑎
1
𝑛𝑛 ∙ 𝑏𝑏
1
𝑛𝑛 Raising product to a power
= √𝑎𝑎
𝑛𝑛
∙ √𝑏𝑏
𝑛𝑛
Rewriting back to radical form
Example 1. Simplify
a. √12 b. √4000 c. √108
3
d.√32
4
Solutions:
a. √12 = √4 ∙ 3 Express 12 as factors where one is the greatest perfect square
= √4 ∙ √3 Apply Product Law of Radicals
= �22 ∙ √3 Replace 𝟒𝟒 by 𝟐𝟐𝟐𝟐
= 2√3
Note that: √𝒂𝒂𝒏𝒏
𝒏𝒏
= 𝒂𝒂
b. √4000 = √400 ∙ 10 c. √108
3
= √27 ∙ 4
3
d.√32
4
= √16 ∙ 2
4
= √400 ∙ √10 = √27
3
∙ √4
3
= √16
4
∙ √2
4
= �202 ∙ √10 = �33
3
∙ √4
3
= �24
4
∙ √2
4
= 20√10 = 3√4
3
= 2√2
4
7. Example 2: Simplify
a. �𝑥𝑥6𝑦𝑦3 b. √−24𝑥𝑥9
3
c. −2𝑛𝑛√81𝑚𝑚8𝑛𝑛12
4
Solutions:
a. �𝑥𝑥6𝑦𝑦3 = �𝑥𝑥3 ∙ 𝑥𝑥3 ∙ 𝑦𝑦2 ∙ 𝑦𝑦 b. √−24𝑥𝑥9
3
= �−8 ∙ 3 ∙ 𝑥𝑥3 ∙ 𝑥𝑥3 ∙ 𝑥𝑥3
3
= �(𝑥𝑥3)2 ∙ 𝑦𝑦2 ∙ 𝑦𝑦 = �(−2)3 ∙ 3 ∙ (𝑥𝑥3)3
3
= �(𝑥𝑥3)2 ∙ �𝑦𝑦2 ∙ �𝑦𝑦 = �(−2)3
3
∙ �(𝑥𝑥3)3
3
∙ √3
3
= 𝑥𝑥3
∙ 𝑦𝑦 ∙ �𝑦𝑦 = −2 ∙ 𝑥𝑥3
∙ √3
3
= 𝑥𝑥3
𝑦𝑦�𝑦𝑦 = −2𝑥𝑥3
√3
3
c. −2𝑛𝑛√81𝑚𝑚8𝑛𝑛11
4
= −2𝑛𝑛�34 ∙ 𝑚𝑚2 ∙ 𝑚𝑚2 ∙ 𝑚𝑚2 ∙ 𝑚𝑚2 ∙ 𝑛𝑛2 ∙ 𝑛𝑛2 ∙ 𝑛𝑛2 ∙ 𝑛𝑛2 ∙ 𝑛𝑛3
4
= −2𝑛𝑛�34 ∙ (𝑚𝑚2)4 ∙ (𝑛𝑛2)4 ∙ 𝑛𝑛3
4
= −2𝑛𝑛�34
4
∙ �(𝑚𝑚2)4
4
∙ �(𝑛𝑛2)4
4
∙ �𝑛𝑛3
4
= −2𝑛𝑛 ∙ 3 ∙ 𝑚𝑚2
∙ 𝑛𝑛2
∙ �𝑛𝑛3
4
= −6𝑚𝑚2
𝑛𝑛3 �𝑛𝑛3
4
Simplifying Radicals within the Radical
If a represent a nonnegative real number and, m and n are positive integer,
�√𝒂𝒂
𝒏𝒏
𝒎𝒎
= √𝒂𝒂
𝒎𝒎𝒎𝒎
Thus, the mth root of the nth root a nonnegative number is equal to 𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛𝑛𝑛ℎ root of that
number.
Proof:
� √𝑎𝑎
𝑛𝑛
𝑚𝑚
= �𝑎𝑎
1
𝑛𝑛�
1
𝑚𝑚 Express 𝒂𝒂 raised to rational exponents
= 𝑎𝑎
1
𝑚𝑚𝑚𝑚 Apply the Power Law of Exponent
= √𝑎𝑎
𝑚𝑚𝑚𝑚
Rewriting back to radical form
Example1: Express in simplest radicals.
a. �√𝑥𝑥 b. �√𝑎𝑎
3
c. ��√𝑚𝑚
3
To watch a video tutorial on Product Rule of Radicals by David McCollister (2013),
visit this link https://youtu.be/rh72wecqFbs
9. Let’s Remember
If a and b represent a nonnegative real number and, m and n are positive integer then the
following properties can be applied to simplify radicals;
√𝒂𝒂𝒂𝒂
𝒏𝒏
= √𝒂𝒂
𝒏𝒏
∙ √𝒃𝒃
𝒏𝒏
√𝒂𝒂𝒏𝒏
𝒏𝒏
= 𝒂𝒂 �√𝒂𝒂
𝒏𝒏
𝒎𝒎
= √𝒂𝒂
𝒎𝒎𝒎𝒎
A radical is said to be in simplest form if the following conditions are satisfied
1. The radicand does not contain factors which are perfect powers of the index.
2. The radicand does not contain fractions.
3. The index of the radicand is the smallest possible number.
4. The denominator is free of radicals
Let’s Apply
Answer the following questions:
1. Can the Product Law of Radicals be applied in simplifying √3
3
∙ √3 ? Why or why not?
2. Explain why √−16 ∙ √−4 = √64 = 8 is an incorrect application of Product Law of Radicals.
3. Why do you think that √𝟓𝟓𝟐𝟐 = 𝟓𝟓 ?
Let’s Evaluate
Directions: Read each question carefully and solve if necessary. Choose the letter of the
correct answer and write it before the number.
1. Rewrite ��√𝑥𝑥 as single radical.
A. √𝑥𝑥
8
B.√𝑥𝑥
6
C. √𝑥𝑥
4
D. √𝑥𝑥
2. Simplify √16𝑥𝑥5
3
A. 4√𝑥𝑥
3
B. 2√2𝑥𝑥
3
C. 2x√2𝑥𝑥2
3
D. 4x
3. Simplify �√3
3
�
6
A. √3
3
B. 18 C. √3
18
D. 9
10. 4. Simplify �32𝑥𝑥5𝑦𝑦2
A. 4𝑥𝑥3
𝑦𝑦√2 B. 4𝑥𝑥2
𝑦𝑦√2 C. 4𝑥𝑥𝑥𝑥√2𝑥𝑥 D.4𝑥𝑥2
𝑦𝑦√2𝑥𝑥
5. Simplify �√64
3
A. 2 B. 4 C. 8 D. 16
6. Which of the following expressions cannot be simplified using Product Law of Radicals?
A. √18 B. √5
3
∙ √15
3
C. 12
1
2 𝐷𝐷. √2 ∙ √2
3
7. Which of the following is true?
A. √50 = 25√2 B.√𝑥𝑥16 = 𝑥𝑥4
C.√7 ∙ √6 = 42 𝐷𝐷. √3𝑎𝑎 ∙ √15𝑏𝑏 = 3√5𝑎𝑎𝑎𝑎
8. What is the greatest perfect square factor of 48?
A. 4 B. 8 C. 16 D. 24
9. What is the simplified form of √8?
A. 2 B. 2√2 C. 4 D. 23
10. The greatest perfect cube factor of 48?
A. 4 B. 8 C. 16 D. 24
https://forms.gle/yqTatFSa6uNRj7sH8
11. References
Nivera, Gladys C., et. al., GRADE 9 MATHEMATICS (Patterns and Practicalities), Salesiana
Books, 2013
Orines, Fernando B., et. al., NEXT CENTURY MATHEMATICS (Intermediate Algebra),
Phoenix Publishing House, 2003
Oronce, Orlando A., et. al., E-MATH 9 (Work text in Mathematics), Rex Book Store, 2015
Sabado, Jennica Alexis B., Taal Volcano [image] [Accessed 6 January 2021]
Development Team of the Module
Writer: JOSEPH C. LAGASCA
Editors:
Content Evaluators: JOEY N. ABERGOS
AMELIA A. CANZANA
ALMA J. CAPUS
JENNIFER N. CONSTANTINO
MARIO D. DE LA CRUZ JR.
NAUMI G. LIGUTAN
DONALYN S. MIÑA
JULIUS PESPES
Language Evaluator: MARICAR G. RAQUIZA
Reviewers: DR. LELINDA H. DE VERA
MIRASOL I. RONGAVILLA
JENNICA ALEXIS B. SABADO
DR. MELEDA POLITA
Layout Artist: JOSEPH C. LAGASCA
Management Team: DR. MARGARITO B. MATERUM, SDS
DR. GEORGE P. TIZON, SGOD-Chief
DR. ELLERY G. QUINTIA, CID-Chief
MRS. MIRASOL I. RONGAVILLA, EPS-Mathematics
DR. DAISY L. MATAAC, EPS-LRMS / ALS
For inquiries, please write or call:
Schools Division of Taguig city and Pateros Upper Bicutan Taguig City
Telefax: 8384251
Email Address: sdo.tapat@deped.gov.ph
