This document provides instruction on multiplying and dividing radical expressions. It begins by stating the objectives of learning how to multiply and divide radicals by applying properties of radicals. It then provides examples of multiplying radicals with the same index, different indices but same radicand, and different indices and radicands. Examples of dividing radicals are also provided. The document demonstrates multiplying radicals by binomials and conjugates. It includes practice problems for students to work through involving multiplying and dividing radical expressions.

1. (Effective Alternative Secondary Education)
MATHEMATICS II
MODULE 5
Radical Expressions
BUREAU OF SECONDARY EDUCATION
Department of Education
DepEd Complex, Meralco Avenue, Pasig City
1
Y
X

2. Module 5
Radical Expressions
What this module is about
Just as you did in the case of adding and subtracting radical expressions,
this module will allow you to multiply and divide them by applying the same basic
procedures in dealing with algebraic expressions. You will constantly be using
properties of radicals which is in the box for easy reference.
What you are expected to learn
1. Recognize basic radical notation
2. apply the basic properties of radicals to obtain an expression in
simplest radical form.
3. multiply and divide radical expressions.
How much do you know
A. Multiply the following expressions..
_ _
1. 4√3 . 3√3
_ _
2. 5√7 . 2√7
_ _
3. 2√5. √7
_ _
4. 5√2 . √5
__ _
5. (2√x2
b)(5√b )
2
__ _ _
Property 1 √ab = √a . √b
_
Property 2. a = √a
b √b

3. B. Divide the following expressions.
_ _
1. √2 ÷ √3
_ _
2. 3
√4 ÷ 3
√6
_ _
3. √2 ÷ 3
√2
_ _
4. √2 ÷ (2 + √3)
__ _ _
5. √xy ÷ (√x - √y)
What will you do
Lesson 1
Multiplication of Radical Expression
In multiplying radical, there are three cases to be considered. These are:
a. Indices are the same. When multiplying radicals having the same
index,
_ _ __
apply: n
√x . n
√y = n
√xy and then if necessary, simplify the resulting
radicand.
b. Indices are different but radicands are the same. To find the product of
radicals with different indices, but the same radicand, apply the
following steps:
1. transform the radical to fractional exponents.
2. multiply the powers by applying: xm
. xn
= xm+n
(law of exponent)
3. rewrite the product as a single radical.
4. simplify the resulting radicand if necessary.
c. Indices and radicands are different. To find the product with different indices
and radicands, follow the following steps:
1. transform the radicals to powers with fractional exponents.
2. change the fractional exponents into similar fractions.
3

4. 3. rewrite the product as a single radical
4. Simplify the resulting radicand if necessary.
Multiplying monomial radicals
Rules to follow:
Rule 1. If radicals to be multiplied have the same indices, follow the steps
in the examples.
_ _ _
Example 1. Multiply: √2.√3.√5
Solution: Write the product of two or more radicals as a single expression.
_ _ _ ____
√2.√3.√5 = √ 2.3.5
__
= √30
__ __
Example 2. Find the product: √12 . √18
Solution: There are two approaches to solve.
__ __ _____
√12 . √18 = √12.18 by property 1
___
= √216 Look for the largest perfect square
factor of 216, which is 36.
__ _
= √36 . √6
_
= 6√6
Second approach: First put each radical into simplest form.
__ __ _ _ _ _
√12 . √18 = √4. √3 . √9. √2
_ _
= 2√3 . 3√2 Rearrange the factors.
_ _
= 2.3√3 √2
_
= 6 √6
Note that the second approach used kept numbers much smaller. The
arithmetic was easier when the radical is simplified first.
_ __
Example 3. Find the product: √7 . √14
Solution: _ __ ____
√7 . √14 = √ 7.14
4

5. __
= √98 express the radicand as product
of the largest perfect square factor.
__ _
= √ 49. √2
_
= 7√2
_ _
Example 4. Multiply: a√3 . b√6
Solution: _ _ ___
a√3 . b√6 = ab√3.6 simply multiply the radicand
having the same index.
__
= ab√18 express the radicand as product
of the largest square factor
_ _
= ab√9 . √2
_
= 3ab√2
___ ____
Example 5. Get the product: √2ab3
. √12ab
Solution: ____ ____ ___________
√2ab3
. √12ab = √ (2ab3
).(12ab) applying the law of
exponent
_____
= √24a2
b4
expressing the radicand
as the largest square
factors
_ _ _ _
= √4 .√6 √a2
√b4
_
= 2ab2
√6
Rule 2. If the radicals have different indices but same radicands, transform the
radicals to powers with fractional exponents, multiply the powers by
applying the multiplication law in exponents and then rewrite the product
as single radical.
_ _
Example 6. √5 . 4
√ 5
_ _
Solution: √5 . 4
√5 = 51/2
. 51/4
= 5 ½ + ¼
5

6. = 53/4
__ ___
= 4
√53
or 4
√125
____ ____
Example 7. (4
√2x – 1) ( 3
√2x – 1
Solution:
_____ _____
(4
√2x – 1 ) ( 3
√2x – 1) = (2x -1 )1/4
(2x – 1)1/3
= (2x – 1) ¼ + 1/3
= (2x – 1) 7/12
_______
= 12
√(2x – 1)7
Rule 3: If radicals have different indices and different radicands, convert the
radicals into powers having similar fraction for exponents and rewrite the
product as a single radical. Simplify the answer if possible.
_ _
Example 8. √2 3
√3
Solution: _ _
√2 3
√3 = 21/2
. 31/3
= 23/6
. 32/6
__ __
= 6
√23
. 6
√32
____
= 6
√ 8 . 9
__
= 6
√72
_ _
Example 9. 4
√2 . 3
√5
Solution: _ _
4
√2 . 3
√5 = 21/4 .
51/3
= 23/12
. 54/12
__ __
= 12
√23
. 12
√54
_ ___
= 12
√8 . 12
√625
_____
= 12
√ 5000
Multiplying a radical by a binomial
6

7. In each of the following multiplication, you are to use the distributive
property to expand the binomial terms.
_ _ _
Example 10. Multiply: √3 ( 2√3 + √5)
Solution: Using the distributive law, then
_ _ _ _ _ _ _
√3 (2√3 + √5) = √3 . 2√3 + √3 . √5
_ _ ____
= 2√3.√3 + √ 3. 5
___
= 2.3 + √15
__
= 6 + √15
_ _ _
Example 11. Multiply and simplify: 2√x (√x - 3) – 4(3 - 5√x)
Solution: Proceed as if there are no radicals- using the distributive law to
remove the parentheses;
_ _ _ _ _ _ _
2√x (√x - 3) – 4(3 - 5√x) = 2√x √x - 6√x – 12 + 20√x
_ _
= 2 x -6√x – 12 + 20√x
_ _
= 2x - 6√x–12 + 20√x combine like terms
_
= 2x + 14√x – 12
Binomial Multiplication.
This method is very much similar to the FOIL method. The terms are
expanded by multiplying each term in the first binomial by each term in the
second binomial.
_ _ _ _
Example 12. (4√3 + √2) (√3 -5√2
_ _ _ _
Solution: (4√3 + √2) (√3 -5√2)
Use the FOIL method, that is multiplying the first terms, outer
terms, inner terms and the last terms.
_ _ _ _ _ _ _ _
= 4(√3)(√3) -4√3(5√2) + √2(√3) - √2(5√2)
_ _ _ _
= 4(√3)2
- 20√6 + √6 -5(√2)2
7

8. _ _
= 4 . 3 - 20√6 + √6 – 5 . 2
_ _
= 12 - 20√6 + √6 – 10
_
= 2 - 19√6
_ _ _ _
Example 13. (√a + √3) (√b + √3)
_ _ _ _
Solution: (√a + √3) (√b + √3) FOIL these binomial then
simplify.
_ _ _ _ _ _ _
= √a√b + √3√a + √3√b + (√3)2
__ __ __
= √ab + √3a + √3b + 3
_ _
Example 14. Multiply and simplify: (√7 - √3 )2
Solution. Watch out! Avoid the temptation to square them
separately.
_ _ _ _ _ _
(√7 - √3)2
= (√7 - √3) (√7 - √3)
_ _ _ _ _ _ _ _
= √7 √7 - √7 √3 - √7 √3 + √3 √3
__ __
= 7 - √21 - √21 + 3 Combine like terms
__
= 10 - 2√21
_ ____
Example 15. (√a – 3)2
– (√a – 3 )2
Solution: Note the difference between the two expressions being
squared.
The first is a binomial; the second is not.
_ ___ _ _ ____ ___
(√a – 3)2
– (√a – 3 )2
= (√a – 3)(√a – 3) - √a – 3 √a-3
_ _ _ _
= √a√a - 3√a -3√a + 9 – (a - 3)
Note that the parentheses around a – 3 is essential.
8
Remember: (a+b)2
≠a2
+ b2

9. _
= a – 6 √a + 9 – a + 3
_
= -6√a + 12
Multiplying Conjugate Binomials
The product of conjugates are always rational numbers. The product of a
pair of conjugates is always a difference of two squares (a2
– b2
), multiplication of
a radical expression by its conjugate results in an expression that is free of
radicals.
__ __
Example 16. (√13 -3) (√13 + 3)
Solution: Multiply out using FOIL.
__ __ __ __ __ __
(√13 -3) (√13 + 3) = √13 √13 + 3√13 - 3√13 – 9 The middle
terms combine to 0.
= 13 – 9
= 4 This answer does
not involve radical.
_ _ _ _
Example 17. (√5 + √7 ) (√5 - √7) A difference of squares
_ _ A square of a root is the
= (√5)2
– (√7)2
original integer
= -2 Simplified
_ _ _ _
Example 18. (√7 + 2√3)(√7 - 2√3)
_ _
= (√7 )2
– (2√3 )2
= 7 – 12
= -5
9

10. Try this out
Perform the indicated operations. Simplify all answers as completely as
possible.
A.
_ __ _ __
1. √3√11 6. √5 √45
_ _ __ _ _ __
2. √3√5√13 7. √2√6√10
_ _ _ _ _
3. √6 √24 8. √3 √5 √6
__ __ __ __
4. √18 √32 9. √24 √28
_ _
5. (-4√2 )2
10. ( 3√5 )2
B.
__ _ _ _ _ _
11. 2√5c . 5√5 16. (2√3 - √7)(2√3 + √7)
_ _ _ ____
12. 2√5 (5√3 + 3√5) 17. ( 1 + √x + 2 )2
_ _ _ _ _
13. (2√5 -4)( 2√5 + 4) 18. √3 ( 2√3 - 3√2)
_ _ _ _ _ _ _ _
14. (3√3 - √2) ( √2 + √3) 19. 3√2 (√2 – 4)+ √2 (5 - √2)
_ _ _ _
15. (√3 + 2) (√3 -5) 20. (√x + √3 )2
C. What’s Message?
Do you feel down with people around you? Don’t feel low. Decode the
message by performing the following radical operations. Write the words
corresponding to the obtained value in the box provided for.
10

11. 11

12. are not
_ _
√2 . 5√8
and irreplaceable
_ _
3
√7 . 4
√7
consider yourself
_ _
4√3 . 3√3
Do not
_ _
√9 . √4
Each one
____ _____
3
√9xy2
. 3 3
√3x4
y6
for people (4√3a3
)2
is unique
_ __
√3 . 3
√18
more or less
__ _
√27 . √3
nor even equal
_ _
√a (√a3
– 7)
of identical quality
_ _
5√7 . 2√7
to others
__ __ ___
(√5a)(√2a)(3√10a2
)
20
12
6
2048a3
36 9 _
a2
-7√a
30a2
_
12
√7
___
9xy2 3
√x2
y2
___
6
√54
70

13. Lesson 2
Division of radicals
Dividing a radical by another radical, follows the rule similar to
multiplication. When a rational expression contains a radical in its denominator,
you often want to find an equivalent expression that does not have a radical in
the denominator. This is rationalization. Study the following examples.
__
Example 1. Simplify: √72
√6
Solution: You are given two solutions:
__
b. Simplify √72. b. Make one radical expression
___ __ __ __
√72 = √36 √2 √72 = 72
√6 √6 √6 6
_ __
= 6√2 Rationalize = √12
√6 _ _
_ _ = √4 √3
= 6√2 . √6 _
√6 √6 = 2√3
__
= 6√12
6
__
= √12
_ _
= √4 √3
_
= 2√3
Note: Clearly the second method is more efficient. If you have the quotient of
two radical expressions and see that there are common factors which can be
reduced, it is usually method 2 is a better strategy, first to make a single radical
and reduce the fraction within the radical sign. then proceed to simplify the
remaining expression.
___
Example 2. √6b7
_
√30ab
___
Solution: √6b7
__ = 6b7
Reduce
13

14. √30ab 30ab
b6
= 5a
_
= √b6
√5a
= b3
√5a
__
= b3
. √5a
√5a √5a
__
= b3
√5a
5a
Rationalizing binomial denominators
The principle used to remove such radicals is the familiar factoring
equation. If a or b is square root, and the denominator is a + b, multiply the
numerator and the denominator by a – b and if a or b is a square root and the
denominator is a + b, multiply the numerator and the denominator by a – b. (a +
b) (a – b) = a2
- b2
Example 3. ___2___
√7 - √5 __ __
Solution: the denominator is √7 - √5, is the difference, so multiply the
numerator and the numerator by the sum √7 + √5:
_ _ _ _
___2___ x √7 + √5 = 2(√7 + √5)
√7 - √5 √7 + √5 (√7)2
– (√5)2
_ _
= 2(√7 + √5 )_
7 – 5
_ _
= 2(√7 + √5) Simplify
2
_ _
= √7 + √5
Example 4. ___20___
√10 + √6
__ _
14

15. Solution: ___20___ = ___20___ . √10 - √6
√10 + √6 √10 + √6 √10 - √6
__ _
= 20(√10 - √6)
10 – 6
__ _
= 20(√10 - √6 )
4
__ __ _ _
= 5(√10 - √6) or 5√10 - 5√6
Example 5. Simplify as completely as possible: ___8___ - 10
3 - √5 √5
Solution: Begin by rationalizing each denominator. Keep in mind that each
fraction has sits own rationalizing factor.
_ _
___8___ - 10 = ___8___ . 3 + √5 - 10 . √5
3 - √5 √5 3 - √5 3 + √5 √5 . √5
_ _
= 8(3 + √5) - 10√5 Reduce each fraction
9 – 5 5
_ _
= 8(3 + √5) - 10√5 Simplify the numerator
4 5 and denominator which
_ _ are not radicand.
= 2(3 + √5) - 2√5 Combine similar
radicands.
_ _
= 6 + 2√5 - 2√5
= 6
__
Example 6. Simplify: 12 + √18
6
Solution: Begin by simplifying the radical.
__ _ _
12 + √18 = 12 +√9 √2
6 6
_
= 12 + 3√2 Factor out the common factor
6 of 3 in the numerator.
_
= 3(4 + √2) simplify
6
15

16. _ _
= 4 + √2 or 2 + √2
2 2
_ _
Example 7. √2 ÷ 3
√2
_ _ _
Solution: √2 ÷ 3
√2 = __√2__
3
√2
= 21/2
Change the radicals to fractional exponent.
21/3
= 23/6
Change the fractional exponents to similar
22/6
fractions
= 6
23
Transform the expression as a single radical.
22
and simplify.
_
= 6
√2
_____ _______
Example 8. Express as a single radical: √4xy2
z2
÷ 6
√16xy2
z4
_____ _______
Solution: √4xy2
z2
÷ 6
√16xy2
z4
Transform to fraction
_____
= __√4xy2
z2
__
6
√16xy2
z4
= (4xy2
z3
)1/2
Change to fractional exponent
(16xy2
z4
)1/6
= (4xy2
z3
)3/6
Change the fractional
(16xy2
z4
)1/6
exponent to similar fractions.
_______
= 6
√(4xy2
z2
)3
Rewrite as radical expressions
6
√16xy2
z4
the radicand to powers.
= 6
64x3
y6
z6
Simplify.
16xy2
z4
= 6
4x2
y4
z2
_ _
Example 9. Perform: √2 ÷ (2 + √3)
_ _ _
Solution: √2 ÷ (2 + √3) = __√2__ rewrite the expression
16

17. 2 + √3
_ _
= __√2__ . 2 -√3 rationalize
2 + √3 2 - √3
_ _
= 2√2 - √6 simplify
4 - 3
_ _
= 2√2 - √6
__ _ _
Example 10. Simplify: √xy ÷ (√x - √y)
__ _ _ __
Solution: √xy ÷ (√x - √y) = __√xy__ rewrite the expression
(√x - √y)
__ _ _
= __√xy__ . √x + √y rationalize
√x - √y √x + √y
___ ___
= √x2
y + √xy2
x – y
_ _
= x √y + y√x
x – y
Try this out
A. Divide and simplify
__ __
1. 6√18 ÷ 12√40
__ __
2. 8√19 ÷ 4√38
_ _
3. 20√3 ÷ 5√3
_ _
4. 42√6 ÷ 3√6
__ _
5. -4√20 ÷ √2
_ _
6. 10√18 ÷ 2√9
__ __
7. 5√96 ÷ 2√24
17

18. __ __
8. 3/7 √30 ÷ 1/3 √15
__ __
9. 20√46 ÷ 5√23
_ __
10. 6√3 ÷ √18
_ __
11. 12√2 ÷ 2√27
_ __
12. 12√6 ÷ ¼ √72
__ ___
13. √50 ÷ √125
__ ___
14. √45 ÷ √400
15. 3
3x2
b ÷ 4
25xy2
B. Simplify
__
1. √10 ÷ 3
2
_
2. 3
3 ÷ √3
3. 4
3 ÷ 3
3
4. 3
6 ÷ 4
6
5. 3
36 ÷ 4
6
_ _
6. √9 ÷ √3
7. 4
27 ÷ 3
2
8. __1__
2 + √5
9. __1__
3 - √11
10. __3__
√3 – 1
D. Why is tennis a noisy game?
18

19. Solve the radicals by performing the indicated operation. Find the answer below
and exchange it for each radical letter.
19

20. _ _
√2 ÷ √3
_ _
3
√8 ÷ 3
√6
__ _
4
√36 ÷ 4
√6
_ _
3
√4 ÷ 3
√6
___7___
√6 + √5
_ __
4√6 - 3√21
√3
_ _
√2 ÷ 3
√2
_ __
√5 ÷ √15
____ ____
3
√ 3x2
b ÷ 3
√25xy2
_ _
2
√2 ÷(2+√3) _1_
√x
___ _
3
√108 ÷ 3
√2
__ __
5√63 ÷ 6√7 400
20
_1_
√5
__
6√28
3√4
__ _
√80 ÷ √5
__
20√46
5√23
__1__
2+√5
_ _
10√18 ÷ 2√9
__ __
5√96 ÷ 2√24
6 5
__3__
√3 - 1
__ ___
√25 ÷ √625
_ _
3
√3 ÷ 3
√5
√6
3
√x
2 7 4
_
5√2
3
√36
3 6
√2 5
2
_______
3
√15bxy
5y
_
√5
5
_
√5
5
_
4
√ 6
3√3+
3
2
3
√18
3 2
_
2√7
3
√75
5
_
4√2
_
-2+√5
_
2√5
√3
3
_ _
4√7-3√7
_
3 3
√2
25
6
Let us summarize
20
E
I
L
S
A
E
R
A
RP
KE
C Y
V E
RA
R A
Y T
SE
Definition:
The pairs of expressions like x - √y and x + √y or √x - √y and √x + √y
are called conjugates. The product of a pair of conjugates has no radicals in it.
Hence, when we rationalize a denominator that has two terms where one or more
of them involve a square-root radical, we multiply by an expression equal 1, that
is, by using the conjugate of the denominator.

21. What have you learned
A. Fill in the blanks.
1. For a = b2
, _______is the square root of ______.
2. When no index is indicated in a radical, then it is understood that the index
is _____.
3. In radical form, 169 3/2
is written as ____ or ____
__
4. In simplest form. √54 is ____
__
5. In simplest form 3
√16 is ____
__
6. In simplest form 4
√64 is ____
__
7. in simplest form, 6
√16 is ___
_____
8. In simplest form √50x7
y11
____ ____
9. The product of (3√ 2 + 4)(3√2 – 4)
__ __
10. The product √26
. 4
√4 _ __ __
11. The combined form 5√7 -2√28 - 3√48 is ___________.
__
12. In simplest form, the quotient √27 = _______
√48
___
13. In simplest , the quotient 3
√135 = _____
3
√40 _
14. In simplest form, the quotient __√7__
√3 - √2
______
15. In simplest form, the quotient 4
√162x6
y7
= ____
4
32x8
y
21

22. Answer Key
How much do you know:
A. 1. 3.4.3 = 36
2. 5.2.7 = 70
__
3. 2√35
4.5√10
5. 10bx
_
B. 1. √6/3
__
2. 3
√18/3
_
3. 6
√2
_ _
4. 2√2 - √6
_ _
5. x√y + y√x
x-y
Try this out
Lesson 1
__
A. 1. √33
___
2. √199
3. 12
__
4. 12√12
5. 32
6. 15
_
7. 2√3
__
8. 3√10
__
9. 2√42
22

23. 10.45
_
B. 11. 50√c
__
12. 10√15 + 30
13. 4
_ _
14. 2√6 + 7 or 7 + 2√6
_ _
15. -3√3 – 7 or -7 - 3√3
16. 5
____
17. 3 + x + 2 √x + 2
_
18. 6 - 3√6
_
19. 4 - 7√2
__
20. 3 + x + 2√3x
C.
23
Do not consider
yourself
more or less nor even
equal
to others
for people are not of identical
quality
each one Is unique
and are
irreplaceable

24. Lesson 2
Try this out.
_ _
A. 1. 3√5 11. 2√6
20 3
__ __
2. 2√19 12. √12
19 4
__
3. 4 13. √10
5
_
4. 14 14. 3√5
__ _____
5. - 4√10 15. 4
√75bxy2
_ 5y
6. 5 √2
7. 10
_
8. 9√2
2
_
9. 4√2
_
10. √6
___ _
B. 1. 6
√250 8. -2 +√5
___ __
2. 6
√243 9. 3 + √11
3 -2
__ _
3. 12
√311
10. 1 + 3√3
3 2
_
4. 12
√6
__ ____
5. 12
√65
or 12
√7776
_
24

25. 6. √3
___
7. 12
√243
C. Why is tennis a noisy game?
√6
3
√x
2 7 4
_
5√2
3
√36
3
_
6
√2
5
6
3
√15xby
5y
√5
5
√5
5
E V E R Y P L A Y E R
R A I S E S A R A C K E T
_
4
√6
3√3+
3
2
3
√18
3 2 2√7
3
√75
5
_
4√2 -2+√5
_
2√5
√3
3
_ _
4√2-3√7
_
3 3
√2
25
6
What have you learned
A. 1. a,b
2. 2
____ ________
3. √169 3
or √ 4826809
_
4. 3√6
_
5. 2 3
√ 2
_
6. 2 4
√4
__
7. 6
√64
___
8. 5x3
y5
√2xy
9. 18 + 9x
10. 6
√16
_ _
11. √7 - 12√3
12. 3/4
13. 3/2
__ __
14. √21 + √14
25

26. ______
15. 34
√72x2
y6
4x
26