9
Mathematics
Quarter 2 – Module 7
Simplifying Radical Expressions
Using the Laws of Radicals
CO_Q2_Mathematics 9 _ Module 7
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. 𝑥 5
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
1
CO_Q2_Mathematics 9 _ Module 7
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 c. 96
b. 24 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
3
10. The simplest form of √40𝑥 8 𝑦 9 𝑧10 is
3
a. 2𝑥 2 𝑦 3 𝑧 3 √5𝑥 2 𝑧
3
b. 3𝑥 2 𝑦 3 𝑧 3 √5𝑥 2 𝑧
c. 2𝑥𝑦𝑧 3√5𝑥𝑦𝑧
d. 3𝑥𝑦𝑧 3√5𝑥𝑦𝑧
121
11. Simplify √ 49
11
a. 7
3
b.
7
11
c. 9
1
d. 7
28𝑥 3𝑦 2 𝑧
12. Simplify the expression √ 7𝑥𝑦 2
a. 4√𝑥𝑧
b. 2𝑥 √𝑧
c. 2√𝑥𝑧
d. 4𝑥 √𝑧
2
CO_Q2_Mathematics 9 _ Module 7
4
13. Rationalize the denominator
3√2𝑥
2√2𝑥
a. 3𝑥
2
b.
√𝑥
2
c. 𝑥
d. 2x
2 3
14. Simplify √ √𝑥 24
a. √𝑥
b. 𝑥 4
c. 𝑥√𝑥
d. 12√𝑥
5 32
15. What is the simplified form of √ 𝑥 ?
5
√𝑥
a. 𝑥
5
2 √𝑥 4
b. 𝑥
c. 2x
d. X
3
CO_Q2_Mathematics 9 _ Module 7
Lesson SIMPLIFYING EXPRESSIONS WITH RATIONAL
1 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.
3
1. √1 6.√8
3
2. ±√49 7.√−64
3
3. −√81 8.− √108
2 3
4. √50 9.√32
3
5. √242 10. √−1
Do you still remember the Laws of Radicals?
6 5
If we have ( 5√𝑥 ) this means, it is also equivalent to √𝑥 6 .
How about √3 ⋅ √𝑦? We may also write it as √3𝑦. It has a somewhat same idea
√3 3
when we have then it can also be written as √𝑦.
√𝑦
3
While we also know that when the index and the exponent is the same, √𝑧 3 then
we can simply write it as 𝑧.
This time, we will use the laws of radicals we learned in simplifying radicals.
4
CO_Q2_Mathematics 9 _ Module 7
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? − √
√− − √−
S O G T A
√ − √
U E I M N
√ − − √ − √
5
CO_Q2_Mathematics 9 _ Module 7
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
What is It
(Communication and Critical Thinking)
SIMPLIFYING RADICAL EXPRESSIONS
Let us study the examples below about simplifying radical expressions
Examples:
Simplify each expression.
4
1. √a4 = 𝐚𝟐 𝑠𝑖𝑛𝑐𝑒 a2 = a2
𝑎𝑛𝑑 a2 • a2 = a4
1
2. √4a2 b 4 c 6 = (4a2 b4 c 6 )2
2 2 4 6
= 22 a2 b 2 c 2
= 𝟐𝒂𝒃𝟐 𝒄𝟑
(note that 4 = 22 )
3. 3
√−54𝑥 4 𝑦 8
3
= √(−27)(2)𝑥 3 • 𝑥 • 𝑦 6 • 𝑦 2
3 3
= √(−3)3 𝑥 3 (𝑦 2 )3 • √2𝑥𝑦 2 Factor out the perfect cube.
𝟑
= −𝟑𝒙𝒚𝟐 √𝟐𝒙𝒚𝟐 Simplify the perfect root.
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CO_Q2_Mathematics 9 _ Module 7
INVESTIGATE 2
Make me SIMPLE
DIRECTION: Write each radical in simplest form.
9 3 8
1. √ 6. √
16 27
3
√25 √3
2. 7. − 3
√100 √5
4 3 7
3. −√5 8. √6
2 3
4. 9. 3
√2 √2
1 3
5. − 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.
75
1. √36
75 √75
Solution: √36 = By Quotient Law of Radicals
√36
√25√3
= 6
By the Multiplication Law of Radicals
𝟓√𝟑
= 𝟔
Since √25 = 5
18x3
2. √49xy2
18x3 18𝑥 2
Solution: √49xy2 = √49𝑦2 Simplify
√18𝑥 2
= By Quotient Law of Radicals
√49y2
√9x2 •2
= By the Multiplication property
√49y2
3x√2
= 7y
Since √9x 2 = 3x and √49y 2 = 7y
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CO_Q2_Mathematics 9 _ Module 7
1 1
What if we are asked to simplify or √2 ?
√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.
1 x2
1. √2b3
6. 3
√x
2 5 𝟐𝐚
2. √3𝑏 7. 𝟑
√𝐚
2√5r 8
3. √m3
8. 3
√3x2
√32c5 d3 8
4. 9.3
√2ab √3x2
3
6√45y3 √5z7 b4
5. 3√5x
10. 3
√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?
Rationalizing the Denominator
8
CO_Q2_Mathematics 9 _ Module 7
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.
√5
Multiply by since √5 • √5 = √25 = 5
√5
1 1 √5 √5 √5
1. = • = The quotient
5
has been rationalized.
√5 √5 √5 √25
Write the numerator and denominator as two separate square roots
√𝟓
= using the Quotient Rule for Radicals
𝟓
To rationalize the denominator of a fraction containing a square root,
7 √7
2. √18 = simply multiply both the numerator and denominator by the
√18
denominator over itself
√7 √18
= • Be sure to simplify the radical in the numerator completely by removing
√18 √18 any factors that are perfect squares.
√126 √9 • √14
= = Be sure to also simplify the fraction by cancelling any common factors
18 18 between the numerator and denominator.
3 • √14 3 • √14
= = The final answer should not contain any radicals in the denominator.
18 3 ⋅ 6
Also, any radicals in the numerator should be simplified completely. And
the fraction should be simplified as well.
√𝟏𝟒
=
𝟔 √2𝑥
Multiply by since √2𝑥 • √2𝑥 = √4𝑥 2 = 2𝑥.
√2𝑥
4 4 √2x
3. = 3√2x •
3√2x √2x
Multiply 2x and 3
4√2x
4√2x 4√2x
= = =
3√4x 2 3(2x) 6x
Remove common factors of 4 and 6
𝟐√𝟐𝐱
= 4
√𝑥 3
𝟑𝐱 Multiply by 4
4 4 4
since √𝑥 • √𝑥 3 = √𝑥 4 = 𝑥
√𝑥 3
4
5𝑦 5𝑦 √𝑥 3
4. 4 = 4 4
√𝑥 √𝑥 √𝑥 3 Simply both the numerator and denominator.
4 4
5𝑦√𝑥 3 5𝑦√𝑥 3
= = The denominator is a binomial, therefore we multiply it by its conjugate.
4
√𝑥 4 𝑥 The conjugate of a binomial is a binomial of the same terms but of
different sign of its second term.
3 3 𝑦−2
5. = √𝑦−2
√𝑦+2 √𝑦+2 √ In this case, the conjugate of √𝑦 + 2 is √𝑦 − 2.
3√𝑦 − 6 3√𝑦 − 6
= =
√𝑦 2 −4 𝑦−4 Multiply the numerators then denominators and simplify. Multiplying the
conjugates would result to √𝑦 2 + 0 + 0 − 4 or simply 𝑦 − 4. Which gives us a
binomial without the radical sign on both terms.
9
CO_Q2_Mathematics 9 _ Module 7
What’s More
A. Simplify each radical.
3
1. √𝑥 5 6. √125𝑥 5 𝑦18
3 3
2. √𝑏7 7. √56𝑚10 𝑛12
2 3
3. √45𝑥 3 𝑦 8 8. √−16𝑥 6 𝑦 3 𝑧12
4
4. √60𝑥 4 𝑦 5 9. √16𝑥 12 𝑦 21
4
5. −2𝑥√3𝑥𝑦 3 𝑧 7 10. √81𝑝25 𝑞37
B. Simplify.
1 2 √11
1. 5. √ 9.
√25 121 −√36
1 4 −3
2. √
49
6. √
9
10. −√9
5 −√36
3. √ 7.
4 √64
3 16
4. 8. −√144
√49
C. Simplify by rationalizing the denominator.
𝟐𝒙 𝟓𝒎 𝟑𝒎
1.√ 𝒚 5.√
𝟕𝒏𝟐
9.√
𝟐𝒏
𝟏 √𝟖𝒙 𝟓
2.√𝟑𝒂𝒃 6. 10.
√𝟑𝒚 √𝟏𝟖𝒚
𝟑√𝟑 𝟐𝒙
3.√𝟔𝒎 7.√𝟓𝒚
𝟒√𝟐 √𝟓𝒂
𝟒. 8.
√𝟐𝟎𝒑 √𝒃
10
CO_Q2_Mathematics 9 _ Module 7
What I Have Learned
Simplifying Radicals
a. Removing Perfect nth Powers √ = √ .√
b.Reducing the index to the lowest possible order √ √ = √ = √ √
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
conjugate of a binomial is a binomial of the same terms but of different
What I Can Do
DIRECTION:
Draw a line to match Column A with the correct answer in Column B.
A B
𝟏. √𝟑𝟑 𝟔𝟑 a.−𝟐𝐱 𝟏𝟖
𝟑 𝟓
𝟐. √𝟐𝟒 𝟓𝟓 𝟐 √ 𝐱𝟒
b. 𝐱
3.√𝟔𝟑𝐦𝟐 c.
𝟐√𝟏𝟓
𝟓
𝟑
4.√−𝟒𝟎𝐱 𝟗 𝐲 𝟏𝟎 d. −𝟐𝐱 𝟑 𝐲 𝟑 𝟑√𝟓𝐲
𝟑
5. −𝟐𝐱√𝟑𝐱𝐲 𝟑 𝐳 𝟕 e. 𝟏𝟎√𝟓𝟎
6.𝟓√𝟑
𝟔√𝟓 f. 𝟓𝟒√𝟐
𝟒
7. 𝟒 g. 𝟑𝐦√𝟕
√𝟖𝐦𝟐
𝟓 𝟑𝟐 h. −𝟐𝐱𝐲𝐳 𝟑 √𝟑𝐱𝐲𝐳
8.√ 𝒙
𝟒
𝟒 𝟐 √𝟐𝐦𝟐
9. 𝟓𝒙𝟒 𝒚𝟐 √ 𝒙𝟐𝟎 𝒚𝟑𝟎 i.
𝟐𝟓 𝐦
𝟑 −𝟏 j. 𝟐𝐱 𝐲 𝟏𝟒 𝟏𝟕
10. 𝟐𝟎𝒙𝟔 √𝟏𝟎𝟎𝟎 𝒙𝟑𝟔
11
CO_Q2_Mathematics 9 _ Module 7
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
3. Find the root of √8
a. 3
b. 4
c. ±2
d. 2
3
4. Which of the following is equal to √−729
a. -9
b. 27
c. 81
d. ±9
3 1
5. √8 when simplify is equal to
1
a.
4
b. 2
1
c. 3
√2
1
d.
2
6. √49𝑥 8 in simplified form
a. a√24.5
b. 7x8
c. 7x4
d. 7√5𝑥 8
7. Simplify √250𝑥 4 𝑦 5
a. 𝑥𝑦√125
b. 5√10𝑥 4 𝑦 5
c. 5x2y2√10𝑦
d. 5√10𝑥 4 𝑦 5
8. Evaluate the expression √160𝑎 5 𝑏4
a. ab√80
b. 4√10𝑎5 𝑏4
c. 4√10𝑎5 𝑏4
d. 4a2b2√10𝑎
12
CO_Q2_Mathematics 9 _ Module 7
3
9. Which of the following is equal with √ √64 ?
6
a. √64
3
b. √64
2
c. √64
2
d. √8
10. It is the process of eliminating the radicals in the denominator of a fraction.
a. Elimination
b. Rationalization
c. Fractionalization
d. Simplification
𝑎 4𝑏 4
11. Simplify √ 𝑐3
.
𝑎 2𝑏 2
a. √ 2
.
𝑎 2𝑏 2
b. √𝑐
𝑐2
𝑎 2𝑏 2
c. 𝑐
√𝑐
𝑎 2𝑏 2
d. 𝑐2
√𝑐 2
8
12. Which of the following is equal to √36𝑎12 𝑏6 ?
3
a. √6𝑎𝑏3
4
b. 𝑎 √6𝑎𝑏3
4
c. 𝑎 √6𝑎 2 𝑏3
4
13. 𝑎 √6𝑎𝑏4 Which of the following is true?
3
3 3 √3𝑥 2
a. √ =
𝑥 𝑥
3
3 3 √𝑥 2
b. √𝑥 = 𝑥
3
3 3 √3𝑥
c. √𝑥 = 𝑥2
2
3 3 √3𝑥 2
d. √ =
𝑥 𝑥
3
14. Which of the following is equal to 5 ?
√2
5
3 √16
a. 2
5
3 √8
b. 2
5
3 √2
c.
16
5
3 √4
d. 2
.15. Which of the following needs to be rationalized?
𝑎2 𝑏2 𝑎𝑏
a. √𝑐 c. √𝑐
𝑐2 𝑐2
𝑎 2 𝑏 2 √𝑐 𝑎2 𝑏2
b. d.
𝑐 √𝑐
13
CO_Q2_Mathematics 9 _ Module 7
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:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
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)
14
CO_Q2_Mathematics 9 _ Module 7
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
√18𝑥 3 𝑦
sides. One of the sides measures .
√2𝑥 4𝑦 2
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?
√18𝑥 3𝑦
5. Find the simplified form of ?
√2𝑥 4𝑦 2
15
CO_Q2_Mathematics 9 _ Module 7
