This module discusses radical expressions and how to simplify them. It covers identifying the radicand and index of radical expressions, simplifying radicals by removing perfect nth roots from the radicand, and rationalizing denominators by removing radicals. The module is designed to teach students to simplify radical expressions, rationalize fractions with radical denominators, and identify radicands and indexes. It provides examples and exercises for students to practice these skills.

1. (Effective Alternative Secondary Education)
MATHEMATICS II
MODULE 3
Radical Expressions
BUREAU OF SECONDARY EDUCATION
Department of Education
DepEd Complex, Meralco Avenue, Pasig City
Y
X

2. Module 3
Radical Expressions
What this module is about
In module 1, you have learned how to identify perfect squares and perfect
cubes. The following lessons will help you simplify radical expressions that are
not perfect nth root and those whose denominators contain square roots. You
will also find the importance of understanding the relation between the radicand,
the root and the index. Learn to appreciate every discussion in this module and
feel free to browse the two previous modules on radical expressions for
clarifications.
What you are expected to learn
This module is designed for you to:
1. Identify the radicand and index in a radical expression.
2. Simplify the radical expression n
x in such a way that the radicand
contains no perfect nth
root.
3. Rationalize a fraction whose denominator contains square roots.
How much do you know
A. Identify the radicand and index of the following and then simplify.
1) 24 6) 3
40
2) 40 7) 3
32
3) 48 8) 4
48
4) 80 9) 5
96
2

3. 5) 63 10) 6
192
B. Simplify the following radical expressions.
1) 32
y125x 6) 11
128b
2) 53
y12a 7) 79
y50x
3) 20mn 8) 4 1585
cb48a
4) 3 6
54a 9) 5 206
b96a
5) 3 12
81x 10) ( )3 5
yx8 +
C. Rationalize the denominator.
1)
2
3
6)
3
1
2)
3
4
7)
4
3
3)
8
5
8) 3
4
1
4)
5
3
9)
3
2
5)
8
2
10)
5
12
What you will do
Lesson 1
Simplifying Radicals
In the radical expression n
x “read as nth root of x, x is called the
radicand, n is the index and the symbol is called radical sign. In 4
16 , 16 is
called the radicand and 4 is the index of the radical. In a , the index is omitted,
it is understood to be 2 or square root.
3

4. Write the radicand as a product of a perfect
square and factors that do not contain perfect
squares.
Examples:
1. 3
27 You are taking the cube root of 27. 27 is the radicand and 3
is the index.
2. 6
64 You are taking the 6th
root of 64. 64 is the radicand and 6 is
the index.
Simplifying Radicals:
A radical is in its simplest form when:
1) There is no perfect nth
power in the radicand when the index is n.
2) There is no radical in the denominator or a fraction in the radicand.
3) The index is the lowest possible index.
In general, the properties of radicals which can be useful in simplifying
radicals can be expressed as follows.
For any nonnegative real numbers x and y and positive integers m and n,
1) nnn yxxy •= The Product Rule for Radicals
2) n
n
n
y
x
y
x
= The Quotient Rule for Radicals
3)
n
1
m
1
xxn m




= by definition of nth
root
mn
1
x= by the Law of Exponents for Powers
mn
x= by definition of nth
root.
Therefore, mnn m
xx =
Examples: Simplify the following:
1. 96 5. 7
x
2. 8 6. 133
y8x3x
3. 5
96 7. ( )2
2x25 +
4. 3 90
Solutions:
1) 96
61696 •=
4
Extract the root of the perfect square factor and
affix the factor which is not a perfect square.

5. 64=
Therefore, the simplest form of 96 is 64 .
2) 8 Factor such that one factor is a perfect square.
248 •= 4 is the perfect square.
22= Simplify.
Therefore, the simplest form of 8 22= .
3) 5
96
5
96 = 55
332 •
= 5
32
Therefore, the simplest form of 5
96 is 5
32 .
4) 3 90
1093903 •=
1033 ••=
109=
Therefore, the simplest form of 3 90 is 109 .
5) 7
x
xxx 67
•=
xx6
•=
xx3
=
Therefore, the simplest form of 7
x is xx3
.
6) 133
y8x3x
( )2xyy4x3xy8x3x 122133
=
2xyy4x3x 122
•=
2xy2xy3x 6
•=
5
Factor
5
32 is 2 because 25
is 32.
Factor such that one factor is a perfect square.
9 is the perfect square.
Simplify.
Write x7
as the product of a perfect square and x.
Use the Product Property of Square Roots.
Simplify the perfect square.
Write the radicand as a factor of a perfect square
and 2xy.
Use the Product Property of Square Roots.
Simplify.

6. 2xyy6x 62
=
Therefore, the simplest form of 133
y8x3x is 2xyy6x 62
.
7) ( )2
2x25 +
( )2
2x25 + ( )2
2x25 +•=
( )2x5 +•=
105x +=
Therefore, the simplest form of ( )2
2x25 + is 5x + 10.
Try this out
A. Identify the radicand and index of the following and then simplify:
1) 16 6) 5
100000
2) 3
64 7) 225
3) 49 8) 4
625
4) 144 9)
81
36
5) 121 10)
256
169
B. Simplify the following:
1) 45 6) 126
2) 12 7) 112
3) 75 8) 180
4) 48 9) 3
40
5) 72 10) 3
32
C. Math Integration:
6
Write the radicand as a product of a perfect
squares. Both factors are perfect squares.
Simplify.

7. Principal Language of Bangladesh
What is the principal language of Bangladesh? This is used by over
210,000,000 natives, and spoken primarily in Bangladesh and India.
To find out, simplify the following radicals. Cross out the boxes that
contain an answer. The remaining boxes will spell out the Bangladesh’s principal
language.
1) 15
b 5) 3 2
80m
2) 19
y 6) 103
b18a2a
3) 5
24x 7) ( )2
5x16 +
4) 7
45b
B
yy9
I
2 3 2
10m
A
5b3b3
B
2
A
2ab6a 52
E
53
N M
bb7
G
x6
L
204x +
A
10y
L
35
I
2x6x2
E
6x2x2
Answer: ____ ____ ____ ____ ____ ____ ____
Source: Math Journal
Volume X – Number 4
D. Math Activity
What is the theme song of the Walt Disney’s animated movie Aladdin
sang by Lea Salonga and Brad King? Simplify each expression below and find
your answer at the bottom. Place the letter in the box above the answer.
4b
3a
xy
4
1
2
3
b
2
8x
5d
4y
3x
4
t7r6
tr
d
4
2
n3m
2a
2
2
3n
m
2
cb5a 42
43
nm
bc
27
3
2
t5r
3c
6rt •
5t
2s2
55
n3m
5
x
4b
3
7
E
D E
W
E
N
E
O
H
AA
E
N
O L
L R
W
WL

8. xya
2
1
2 2
2
xy
a
9y
4
b6a
5c
2
5
Lesson 2
Rationalizing the Denominator of Radicals
A radical is not considered simplified if there is a radical sign in the
denominator. To remove the radical sign is to rationalize.
The process of eliminating the radicals in the denominator is called
rationalization.
Examples:
1)
2
5
To simplify, rationalize or remove the radical sign in the denominator.
Solution:
2
2
2
5
•
4
10
=
2
10
=
Therefore, the simplest form of
2
5
is
2
10
.
8
This is not simplified because there is a radical sign
in the denominator.
Multiply both numerator and denominator by 2 (the
given denominator) to make the denominator a perfect
square.
Simplify.
The denominator is now free
of radicals.
xy
ax
4
10d
5
5x
5
suot
3n
3mn
3n
m4a
2b
3b
2
xy3
2
10cab2
2
10a
tr
td
2
2
7tr3
3
3b
22
nm
bcm
3y
y4
4y
3xy
2b
3ab

9. 2) Simplify: 6
2
z
4x
6
2
6
2
z
4x
z
4x
=
3
z
2x
=
Therefore, the simplest form of 6
2
z
4x
is 3
z
2x
.
3) Simplify:
3
2
3
3
3
2
3
2
•=
9
32
=
3
32
=
Therefore,
3
2
is in simplest form,
3
32
, because no radical remains in
the denominator and the numerator radical contains no perfect-square factors
other than 1.
When the denominator contains a binomial radical expression, simplify the
radical expression by multiplying the numerator and denominator by the
conjugate of the denominator.
4) Simplify:
3y
2y
+
3y
3y
3y
2y
3y
2y
−
−
•
+
=
+
9y
2y32y
2
2
−
−
=
9
Rewrite the radical expression as the
quotient of the square roots.
Simplify.
Multiply the expression by
3
3
to make the denominator
a perfect square.
The denominator is a perfect
square.
Simplify.
Multiply the numerator and denominator by
3y − , the conjugate of 3y + .
Simplify by multiplying both numerator and write it
over the product of the denominator.

10. 9y
2y32y
−
−
= The denominator is now free of radicals.
Try this out
A. Math Integration
An African Festival
This seven-day African festival is celebrated beginning December 26. It
celebrates seven virtues: unity, self-determination, collective work, responsibility,
cooperative economics, creativity, and faith. This celebration means first fruits in
Swahili, and African language.
What is the festival called?
To find out, rationalize the denominator of the following radicals. Encircle
the correct answer. Then write the letter in the blank that goes with the number.
1)
8
3
J
3
2
K
4
6
2)
3
10
W
3
30
X
3
10
3)
3
1
A 3 E
3
3
4)
4
3
M
2
3
N
3
2
5)
8
2
Y
4
2
Z
2
1
6)
5
3
A
5
15
O
3
5
7)
3
5
A
3
35
E
3
3
Answer: ____ ____ ____ ____ ____ ____ ____
1 2 3 4 5 6 7
10

11. Source: Math Journal, Volume X, Number 3
SY 2002-2003
B. Rationalize the denominator.
1)
2
a
6) 3
44
c
ba
2)
5
2c
7) 3
2y
7x
3) 3y
5x
8)
6
y
4)
2x
5mn
9)
x
3
5) 3
2
y
7m
10) 3
16
2d
C. Math Game: (Source: Math Journal, Volume XI, November 29, 2004)
What Did the Farmer Get When He Tried to Reach the Beehive?
To find out, simplify the following radicals. Encircle the letter that
corresponds to the correct answer. Then fill in the blanks below. Have fun!
1)
5
1
Y 55 Z
5
5
2)
8
3
O
2
6
U
4
6
3)
27
5
A
9
15
E
3
5
4)
3
5
S
3
35
R
3
15
5)
7
2
N
7
14
P
7
2
6)
3
52
G
3
152
H
3
10
7)
2
64
A 34 E 24
11

12. 8) 3
4
1
Y 3
4 Z
2
23
9) 3
1000
7
− B
10
73
− C 3
70−
10) 3
25
1
− L
5
53
−
M 3
70−
11) 3
36
5
Y
6
303
Z
5
363
12)
5
7
E
5
35
I
5
57
Answer: ___ __ __ __ __ __ __ __ __ __ __ __
3 9 2 1 8 11 4 12 6 5 7 10
Let’s summarize
Expressions using the radical sign are called radical expressions. In
the expression n
x , n is called the index, x is the radicand and is the radical
symbol. The index of the radical symbol is understood as 2 and is read as
square root.
A radicand is the number inside the radical sign or the number whose root
is being considered.
An index is a small number or letter which indicates the order of the
radical.
A radical is in its simplest form when:
a. There is no perfect nth
power in the radicand when the index is n.
b. There is no radical in the denominator or a fraction in the radicand.
c. The index is the lowest possible index.
The process of eliminating the radicals in the denominator is called
rationalization.
12

13. What have you learned
A. Simplify the following.
1) 32 6) 125
2) 50 7) 112
3) 18 8) 175
4) 28 9) 99
5) 108 10) 288
B. Identify the radicand and index of the following and simplify:
1) 3
54 6) 11
99y
2) 4
48 7) 4
98x
3) 5
96 8) 12
88x−
4) 6
192 9) 102
y162x−
5) 252 10) 119
y50x−
C. Simplify the following:
1)
2
32
6)
6
4
2)
5
45
7)
xy
y4x2
3)
2
98
8)
y3x
y4x
4
52
4)
3
48
9)
62
2
+
5)
3a
27a
10)
57
5
−
13

14. Answer Key
How much do you know
A. 1) 62
2) 102
3) 34
4) 54
5) 73
6) 3
52
7) 3
42
8) 4
32
9) 5
32
10) 6
32
B. 1) 5y5xy
2) 3ay2ay2
3) 5mn2
4) 32
23a
5) 34
33x
6) 2b8b5
7) 2xyy5x 34
8) 4 332
3acc2ab
9) 54
3a2ab
10) ( ) ( )3
yxyx2
2
++
C. 1)
2
6
2)
3
34
3)
4
10
14

15. 4)
5
15
5)
2
1
6)
3
3
7)
2
3
8)
2
23
9)
3
6
10)
5
152
Try this out
Lesson 1
A. 1) 4
2) 4
3) 7
4) 12
5) 11
6) 10
7) 15
8) 5
9)
9
6
10)
16
13
B. 1) 53
2) 32
3) 35
4) 34
15

16. 5) 26
6) 143
7) 74
8) 56
9) 3
52
10) 3
42
C. 1) 7
bb
2) yy9
3) 6x2x2
4) 5b3b3
5) 3 2
10m2
6) 2ab6a 52
7) 204x +
Answer: BENGALI
D. Math Activity:
Answer: “A WHOLE NEW WORLD”
Lesson 2
A. 1)
4
6
K
2)
3
30
W
3)
3
3
E
4)
2
3
M
5)
2
1
Z
6)
5
15
A
7)
3
35
A
16

17. Answer: KWEMZAA
B. 1)
2
2a
2)
5
10c
3) 3y
15xy
4)
2x
10mnx
5) 2
y
7ym
6) 2
22
c
cba
7) 2
2y
14xy
8)
6
6y
9)
x
3x
10)
2
d3
C. Math Game
1)
5
5
Z
2)
4
6
U
3)
9
15
A
4)
3
35
S
5)
7
14
N
17

18. 6)
3
152
G
7) 34 A
8)
2
23
Z
9)
10
73
−
B
10)
5
53
−
L
11)
6
303
Y
12)
5
57
I
Answer: A BUZZY SIGNAL
What have you learned
A. 1) 24
2) 25
3) 23
4) 72
5) 36
6) 55
7) 74
8) 75
9) 113
10) 212
B. 1) 3
23
2) 4
32
3) 5
2 3
4) 6
32
5) 76
18

19. 6) 11y3y5
7) 22
7x
8) 222x6
−
9) 29xy5
−
10) 2xyy5x 54
−
C. 1) 4
2) 3
3) 7
4) 4
5) 3
6)
3
62
7) x2
8)
3x
32y2
9) 






 −
−
2
31
10)
44
557 +
19