Algebraic Expressions and Identities - Exercise 6.3 | Mathematics (Maths) Class 8 PDF Download

 Page 1


    
            
               
 
 
 
                    
    
 
 
           
                                                
 
        
                                                             
 
           
  
                                                    
Q u e s t i o n : 1 0
Find each of the following product:
5x
2 
× 4x
3
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices. However, use of these laws are subject to their applicability in the given
expressions.
In the present problem, to perform the multiplication, we can proceed as follows:
5x
2
×4x
3
= (5 ×4)× x
2
×x
3
= 20x
5
                           ( ? a
m
×a
n
= a
m+n
)
Thus, the answer is 20x
5
.
Q u e s t i o n : 1 1
Find each of the following product:
-3a
2
 × 4b
4
S o l u t i o n :
To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
-3a
2
×4b
4
= (-3 ×4)× a
2
×b
4
= -12a
2
b
4
Thus, the answer is -12a
2
b
4
.
Q u e s t i o n : 1 2
Find each of the following product:
(-5xy) × (-3x
2
yz)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
(-5xy)× -3x
2
yz = {(-5)×(-3)}× x ×x
2
×(y ×y)×z = 15 × x
1+2
× y
1+1
×z = 15x
3
y
2
z
( )
( )
( ) ( ) ( ) ( )
Page 2


    
            
               
 
 
 
                    
    
 
 
           
                                                
 
        
                                                             
 
           
  
                                                    
Q u e s t i o n : 1 0
Find each of the following product:
5x
2 
× 4x
3
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices. However, use of these laws are subject to their applicability in the given
expressions.
In the present problem, to perform the multiplication, we can proceed as follows:
5x
2
×4x
3
= (5 ×4)× x
2
×x
3
= 20x
5
                           ( ? a
m
×a
n
= a
m+n
)
Thus, the answer is 20x
5
.
Q u e s t i o n : 1 1
Find each of the following product:
-3a
2
 × 4b
4
S o l u t i o n :
To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
-3a
2
×4b
4
= (-3 ×4)× a
2
×b
4
= -12a
2
b
4
Thus, the answer is -12a
2
b
4
.
Q u e s t i o n : 1 2
Find each of the following product:
(-5xy) × (-3x
2
yz)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
(-5xy)× -3x
2
yz = {(-5)×(-3)}× x ×x
2
×(y ×y)×z = 15 × x
1+2
× y
1+1
×z = 15x
3
y
2
z
( )
( )
( ) ( ) ( ) ( )
Thus, the answer is 15x
3
y
2
z.
Q u e s t i o n : 1 3
Find each of the following product:
1
4
xy ×
2
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the the law of indices, that is, a
m
×a
n
= a
m+n
.
We have:
1
4
xy ×
2
3
x
2
yz
2
=
1
4
×
2
3
× x ×x
2
×(y ×y)×z
2
=
1
4
×
2
3
× x
1+2
× y
1+1
×z
2
=
1
6
x
3
y
2
z
2
Thus, the answer is 
1
6
x
3
y
2
z
2
.
Q u e s t i o n : 1 4
Find each of the following product:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
= -
7
5
×
13
3
× x ×x
2
× y
2
×y × z ×z
2
= -
7
5
×
13
3
× x
1+2
× y
2+1
× z
1+2
= -
91
15
x
3
y
3
x
3
Thus, the answer is -
91
15
x
3
y
3
x
3
.
 
Q u e s t i o n : 1 5
Find each of the following product:
-24
25
x
3
z × -
15
16
xz
2
y
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
24
25
x
3
z × -
15
16
xz
2
y = -
24
25
× -
15
16
× x
3
×x × z ×z
2
×y = -
24
25
× -
15
16
× x
3+1
× z
1+2
×y
=
9
10
x
4
yz
3
                                                           
Thus, the answer is 
9
10
x
4
yz
3
.
Q u e s t i o n : 1 6
Find each of the following product:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
= -
1
27
×
9
2
× a
2
×a
3
× b
2
×b
2
×c
2
= -
1
27
×
9
2
× a
2+3
× b
2+2
×c
2
= -
1
6
a
5
b
4
c
2
Thus, the answer is -
1
6
a
5
b
4
c
2
.
Q u e s t i o n : 1 7
Find each of the following product:
-7xy ×
1
4
x
2
yz
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-7xy)×
1
4
x
2
yz = -7 ×
1
4
× x ×x
2
×(y ×y)×z = -7 ×
1
4
× x
1+2
× y
1+1
×z = -
7
4
x
3
y
2
z
Thus, the answer is -
7
4
x
3
y
2
z.
Q u e s t i o n : 1 8
Find each of the following product:
(7ab) × (-5ab
2
c) × (6abc
2
)
S o l u t i o n :
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) { ( ) ( )}
( ) ( )
{ ( ) ( )}
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
Page 3


    
            
               
 
 
 
                    
    
 
 
           
                                                
 
        
                                                             
 
           
  
                                                    
Q u e s t i o n : 1 0
Find each of the following product:
5x
2 
× 4x
3
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices. However, use of these laws are subject to their applicability in the given
expressions.
In the present problem, to perform the multiplication, we can proceed as follows:
5x
2
×4x
3
= (5 ×4)× x
2
×x
3
= 20x
5
                           ( ? a
m
×a
n
= a
m+n
)
Thus, the answer is 20x
5
.
Q u e s t i o n : 1 1
Find each of the following product:
-3a
2
 × 4b
4
S o l u t i o n :
To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
-3a
2
×4b
4
= (-3 ×4)× a
2
×b
4
= -12a
2
b
4
Thus, the answer is -12a
2
b
4
.
Q u e s t i o n : 1 2
Find each of the following product:
(-5xy) × (-3x
2
yz)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
(-5xy)× -3x
2
yz = {(-5)×(-3)}× x ×x
2
×(y ×y)×z = 15 × x
1+2
× y
1+1
×z = 15x
3
y
2
z
( )
( )
( ) ( ) ( ) ( )
Thus, the answer is 15x
3
y
2
z.
Q u e s t i o n : 1 3
Find each of the following product:
1
4
xy ×
2
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the the law of indices, that is, a
m
×a
n
= a
m+n
.
We have:
1
4
xy ×
2
3
x
2
yz
2
=
1
4
×
2
3
× x ×x
2
×(y ×y)×z
2
=
1
4
×
2
3
× x
1+2
× y
1+1
×z
2
=
1
6
x
3
y
2
z
2
Thus, the answer is 
1
6
x
3
y
2
z
2
.
Q u e s t i o n : 1 4
Find each of the following product:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
= -
7
5
×
13
3
× x ×x
2
× y
2
×y × z ×z
2
= -
7
5
×
13
3
× x
1+2
× y
2+1
× z
1+2
= -
91
15
x
3
y
3
x
3
Thus, the answer is -
91
15
x
3
y
3
x
3
.
 
Q u e s t i o n : 1 5
Find each of the following product:
-24
25
x
3
z × -
15
16
xz
2
y
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
24
25
x
3
z × -
15
16
xz
2
y = -
24
25
× -
15
16
× x
3
×x × z ×z
2
×y = -
24
25
× -
15
16
× x
3+1
× z
1+2
×y
=
9
10
x
4
yz
3
                                                           
Thus, the answer is 
9
10
x
4
yz
3
.
Q u e s t i o n : 1 6
Find each of the following product:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
= -
1
27
×
9
2
× a
2
×a
3
× b
2
×b
2
×c
2
= -
1
27
×
9
2
× a
2+3
× b
2+2
×c
2
= -
1
6
a
5
b
4
c
2
Thus, the answer is -
1
6
a
5
b
4
c
2
.
Q u e s t i o n : 1 7
Find each of the following product:
-7xy ×
1
4
x
2
yz
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-7xy)×
1
4
x
2
yz = -7 ×
1
4
× x ×x
2
×(y ×y)×z = -7 ×
1
4
× x
1+2
× y
1+1
×z = -
7
4
x
3
y
2
z
Thus, the answer is -
7
4
x
3
y
2
z.
Q u e s t i o n : 1 8
Find each of the following product:
(7ab) × (-5ab
2
c) × (6abc
2
)
S o l u t i o n :
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) { ( ) ( )}
( ) ( )
{ ( ) ( )}
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,  a
m
×a
n
= a
m+n
.
We have:
(7ab)× -5ab
2
c × 6abc
2
= {7 ×(-5)×6}×(a ×a ×a)× b ×b
2
×b × c ×c
2
= {7 ×(-5)×6}× a
1+1+1
× b
1+2+1
× c
1+2
= -210a
3
b
4
c
3
Thus, the answer is -210a
3
b
4
c
3
.
Q u e s t i o n : 1 9
Find each of the following product:
(-5a) × (-10a
2
) × (-2a
3
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-5a)× -10a
2
× -2a
3
= {(-5)×(-10)×(-2)}× a ×a
2
×a
3
= {(-5)×(-10)×(-2)}× a
1+2+3
= -100a
6
Thus, the answer is -100a
6
.
Q u e s t i o n : 2 0
Find each of the following product:
(-4x
2
) × (-6xy
2
) × (-3yz
2
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-4x
2
× -6xy
2
× -3yz
2
= {(-4)×(-6)×(-3)}× x
2
×x × y
2
×y ×z
2
= {(-4)×(-6)×(-3)}× x
2+1
× y
2+1
×z
2
= -72x
3
y
3
z
2
Thus, the answer is -72x
3
y
3
z
2
.
Q u e s t i o n : 2 1
Find each of the following product:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
= -
2
7
× -
3
4
× -
14
5
× a
4
×a
2
× b ×b
2
= -
2
7
×
3
4
×
14
5
×a
4+2
×b
1+2
= -
2
7
×
3
4 2
×
14
2 1
5
×a
6
×b
3
= -
3
5
a
6
b
3
Thus, the answer is -
3
5
a
6
b
3
.
Q u e s t i o n : 2 2
Find each of the following product:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c =
7
9
×
15
7
× -
3
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3
7
× -
3
1
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3 1
7
× -
3
1
5
× a
1+1+2
× b
2+1
× c
2+1
= -a
4
b
3
c
3
Thus, the answer is -a
4
b
3
c
3
.
Q u e s t i o n : 2 3
Find each of the following product:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,a
m
×a
n
= a
m+n
.
We have:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu =
4
3
×(-5)×
1
3
× u
2
×u ×u × v ×v ×v
2
× w ×w
2
×w =
4
3
×(-5)×
1
3
× u
2+1+1
× v
1+1+2
× w
1+2+1
= -
20
9
u
4
v
4
w
4
Thus, the answer is -
20
9
u
4
v
4
w
4
.
Q u e s t i o n : 2 4
Find each of the following product:
(0. 5x)×
1
3
xy
2
z
4
× 24x
2
yz
S o l u t i o n :
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) { ( ) ( ) ( )}
( ) ( )
{ ( )}
{ ( ) }
( ) ( ) ( )
( ) ( ) ( ) { ( )}
( ) ( ) ( )
{ ( )}
( ) ( ) ( )
{ ( ) }
( ) ( ) ( )
( )
( )
( )
( )
( )
( ) { }
( ) ( ) ( )
{ }
( ) ( ) ( )
( )
( )
Page 4


    
            
               
 
 
 
                    
    
 
 
           
                                                
 
        
                                                             
 
           
  
                                                    
Q u e s t i o n : 1 0
Find each of the following product:
5x
2 
× 4x
3
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices. However, use of these laws are subject to their applicability in the given
expressions.
In the present problem, to perform the multiplication, we can proceed as follows:
5x
2
×4x
3
= (5 ×4)× x
2
×x
3
= 20x
5
                           ( ? a
m
×a
n
= a
m+n
)
Thus, the answer is 20x
5
.
Q u e s t i o n : 1 1
Find each of the following product:
-3a
2
 × 4b
4
S o l u t i o n :
To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
-3a
2
×4b
4
= (-3 ×4)× a
2
×b
4
= -12a
2
b
4
Thus, the answer is -12a
2
b
4
.
Q u e s t i o n : 1 2
Find each of the following product:
(-5xy) × (-3x
2
yz)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
(-5xy)× -3x
2
yz = {(-5)×(-3)}× x ×x
2
×(y ×y)×z = 15 × x
1+2
× y
1+1
×z = 15x
3
y
2
z
( )
( )
( ) ( ) ( ) ( )
Thus, the answer is 15x
3
y
2
z.
Q u e s t i o n : 1 3
Find each of the following product:
1
4
xy ×
2
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the the law of indices, that is, a
m
×a
n
= a
m+n
.
We have:
1
4
xy ×
2
3
x
2
yz
2
=
1
4
×
2
3
× x ×x
2
×(y ×y)×z
2
=
1
4
×
2
3
× x
1+2
× y
1+1
×z
2
=
1
6
x
3
y
2
z
2
Thus, the answer is 
1
6
x
3
y
2
z
2
.
Q u e s t i o n : 1 4
Find each of the following product:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
= -
7
5
×
13
3
× x ×x
2
× y
2
×y × z ×z
2
= -
7
5
×
13
3
× x
1+2
× y
2+1
× z
1+2
= -
91
15
x
3
y
3
x
3
Thus, the answer is -
91
15
x
3
y
3
x
3
.
 
Q u e s t i o n : 1 5
Find each of the following product:
-24
25
x
3
z × -
15
16
xz
2
y
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
24
25
x
3
z × -
15
16
xz
2
y = -
24
25
× -
15
16
× x
3
×x × z ×z
2
×y = -
24
25
× -
15
16
× x
3+1
× z
1+2
×y
=
9
10
x
4
yz
3
                                                           
Thus, the answer is 
9
10
x
4
yz
3
.
Q u e s t i o n : 1 6
Find each of the following product:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
= -
1
27
×
9
2
× a
2
×a
3
× b
2
×b
2
×c
2
= -
1
27
×
9
2
× a
2+3
× b
2+2
×c
2
= -
1
6
a
5
b
4
c
2
Thus, the answer is -
1
6
a
5
b
4
c
2
.
Q u e s t i o n : 1 7
Find each of the following product:
-7xy ×
1
4
x
2
yz
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-7xy)×
1
4
x
2
yz = -7 ×
1
4
× x ×x
2
×(y ×y)×z = -7 ×
1
4
× x
1+2
× y
1+1
×z = -
7
4
x
3
y
2
z
Thus, the answer is -
7
4
x
3
y
2
z.
Q u e s t i o n : 1 8
Find each of the following product:
(7ab) × (-5ab
2
c) × (6abc
2
)
S o l u t i o n :
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) { ( ) ( )}
( ) ( )
{ ( ) ( )}
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,  a
m
×a
n
= a
m+n
.
We have:
(7ab)× -5ab
2
c × 6abc
2
= {7 ×(-5)×6}×(a ×a ×a)× b ×b
2
×b × c ×c
2
= {7 ×(-5)×6}× a
1+1+1
× b
1+2+1
× c
1+2
= -210a
3
b
4
c
3
Thus, the answer is -210a
3
b
4
c
3
.
Q u e s t i o n : 1 9
Find each of the following product:
(-5a) × (-10a
2
) × (-2a
3
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-5a)× -10a
2
× -2a
3
= {(-5)×(-10)×(-2)}× a ×a
2
×a
3
= {(-5)×(-10)×(-2)}× a
1+2+3
= -100a
6
Thus, the answer is -100a
6
.
Q u e s t i o n : 2 0
Find each of the following product:
(-4x
2
) × (-6xy
2
) × (-3yz
2
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-4x
2
× -6xy
2
× -3yz
2
= {(-4)×(-6)×(-3)}× x
2
×x × y
2
×y ×z
2
= {(-4)×(-6)×(-3)}× x
2+1
× y
2+1
×z
2
= -72x
3
y
3
z
2
Thus, the answer is -72x
3
y
3
z
2
.
Q u e s t i o n : 2 1
Find each of the following product:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
= -
2
7
× -
3
4
× -
14
5
× a
4
×a
2
× b ×b
2
= -
2
7
×
3
4
×
14
5
×a
4+2
×b
1+2
= -
2
7
×
3
4 2
×
14
2 1
5
×a
6
×b
3
= -
3
5
a
6
b
3
Thus, the answer is -
3
5
a
6
b
3
.
Q u e s t i o n : 2 2
Find each of the following product:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c =
7
9
×
15
7
× -
3
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3
7
× -
3
1
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3 1
7
× -
3
1
5
× a
1+1+2
× b
2+1
× c
2+1
= -a
4
b
3
c
3
Thus, the answer is -a
4
b
3
c
3
.
Q u e s t i o n : 2 3
Find each of the following product:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,a
m
×a
n
= a
m+n
.
We have:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu =
4
3
×(-5)×
1
3
× u
2
×u ×u × v ×v ×v
2
× w ×w
2
×w =
4
3
×(-5)×
1
3
× u
2+1+1
× v
1+1+2
× w
1+2+1
= -
20
9
u
4
v
4
w
4
Thus, the answer is -
20
9
u
4
v
4
w
4
.
Q u e s t i o n : 2 4
Find each of the following product:
(0. 5x)×
1
3
xy
2
z
4
× 24x
2
yz
S o l u t i o n :
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) { ( ) ( ) ( )}
( ) ( )
{ ( )}
{ ( ) }
( ) ( ) ( )
( ) ( ) ( ) { ( )}
( ) ( ) ( )
{ ( )}
( ) ( ) ( )
{ ( ) }
( ) ( ) ( )
( )
( )
( )
( )
( )
( ) { }
( ) ( ) ( )
{ }
( ) ( ) ( )
( )
( )
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(0. 5x)×
1
3
xy
2
z
4
× 24x
2
yz = 0. 5 ×
1
3
×24 × x ×x ×x
2
× y
2
×y × z
4
×z = 0. 5 ×
1
3
×24 × x
1+1+2
× y
2+1
× z
4+1
= 4x
4
y
3
z
5
Thus, the answer is 4x
4
y
3
z
5
.
Q u e s t i o n : 2 5
Find each of the following product:
4
3
pq
2
× -
1
4
p
2
r × 16p
2
q
2
r
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
4
3
pq
2
× -
1
4
p
2
r × 16p
2
q
2
r
2
=
4
3
× -
1
4
×16 × p ×p
2
×p
2
× q
2
×q
2
× r×r
2
=
4
3
× -
1
4
×16 × p
1+2+2
× q
2+2
× r
1+2
= -
16
3
p
5
q
4
r
3
Thus, the answer is -
1
3
p
5
q
4
r
3
.
Q u e s t i o n : 2 6
Find each of the following product:
(2.3xy) × (0.1x) × 0.16
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(2. 3xy)×(0. 1x)×(0. 16) = (2. 3 ×0. 1 ×0. 16)×(x ×x)×y = (2. 3 ×0. 1 ×0. 16)× x
1+1
×y = 0. 0368x
2
y
Thus, the answer is 0. 0368x
2
y.
Q u e s t i o n : 2 7
Express each of the following product as a monomials and verify the result in each case for x = 1:
(3x) × (4x) × (-5x)
S o l u t i o n :
We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., ? a
m
×a
n
= a
m+n
.
We have:
(3x)×(4x)×(-5x) = {3 ×4 ×(-5)}×(x ×x ×x) = {3 ×4 ×(-5)}× x
1+1+1
= -60x
3
Substituting x = 1 in LHS, we get:
LHS = (3x)×(4x)×(-5x) = (3 ×1)×(4 ×1)×(-5 ×1) = -60
Putting x = 1 in RHS, we get:
RHS = -60x
3
= -60(1)
3
= -60 ×1 = -60
? LHS = RHS for x = 1; therefore, the result is correct
Thus, the answer is -60x
3
.
Q u e s t i o n : 2 8
Express each of the following product as a monomials and verify the result in each case for x = 1:
(4x
2
) × (-3x) × 
4
5
x
3
S o l u t i o n :
We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., ? a
m
×a
n
= a
m+n
.
We  have:
4x
2
×(-3x)×
4
5
x
3
= 4 ×(-3)×
4
5
× x
2
×x ×x
3
= 4 ×(-3)×
4
5
× x
2+1+3
= -
48
5
x
6
? 4x
2
×(-3x)×
4
5
x
3
= -
48
5
x
6
Substituting x = 1 in LHS, we get:
LHS = 4x
2
×(-3x)×
4
5
x
3
= 4 ×1
2
×(-3 ×1)×
4
5
×1
3
= 4 ×(-3)×
4
5
= -
48
5
Putting x = 1 in RHS, we get:
RHS = -
48
5
x
6
= -
48
5
×1
6
= -
48
5
? LHS = RHS for x = 1; therefore, the result is correct
Thus, the answer is -
48
5
x
6
.
( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
{ ( ) }
( ) ( ) ( )
{ ( ) }
( ) ( ) ( )
( )
( )
( )
( )
( ) { }
( )
{ }
( )
( )
( )
( )
( )
( )
( )
Page 5


    
            
               
 
 
 
                    
    
 
 
           
                                                
 
        
                                                             
 
           
  
                                                    
Q u e s t i o n : 1 0
Find each of the following product:
5x
2 
× 4x
3
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices. However, use of these laws are subject to their applicability in the given
expressions.
In the present problem, to perform the multiplication, we can proceed as follows:
5x
2
×4x
3
= (5 ×4)× x
2
×x
3
= 20x
5
                           ( ? a
m
×a
n
= a
m+n
)
Thus, the answer is 20x
5
.
Q u e s t i o n : 1 1
Find each of the following product:
-3a
2
 × 4b
4
S o l u t i o n :
To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
-3a
2
×4b
4
= (-3 ×4)× a
2
×b
4
= -12a
2
b
4
Thus, the answer is -12a
2
b
4
.
Q u e s t i o n : 1 2
Find each of the following product:
(-5xy) × (-3x
2
yz)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, a
m
×a
n
= a
m+n
, wherever applicable.
We have:
(-5xy)× -3x
2
yz = {(-5)×(-3)}× x ×x
2
×(y ×y)×z = 15 × x
1+2
× y
1+1
×z = 15x
3
y
2
z
( )
( )
( ) ( ) ( ) ( )
Thus, the answer is 15x
3
y
2
z.
Q u e s t i o n : 1 3
Find each of the following product:
1
4
xy ×
2
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the the law of indices, that is, a
m
×a
n
= a
m+n
.
We have:
1
4
xy ×
2
3
x
2
yz
2
=
1
4
×
2
3
× x ×x
2
×(y ×y)×z
2
=
1
4
×
2
3
× x
1+2
× y
1+1
×z
2
=
1
6
x
3
y
2
z
2
Thus, the answer is 
1
6
x
3
y
2
z
2
.
Q u e s t i o n : 1 4
Find each of the following product:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
7
5
xy
2
z ×
13
3
x
2
yz
2
= -
7
5
×
13
3
× x ×x
2
× y
2
×y × z ×z
2
= -
7
5
×
13
3
× x
1+2
× y
2+1
× z
1+2
= -
91
15
x
3
y
3
x
3
Thus, the answer is -
91
15
x
3
y
3
x
3
.
 
Q u e s t i o n : 1 5
Find each of the following product:
-24
25
x
3
z × -
15
16
xz
2
y
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
24
25
x
3
z × -
15
16
xz
2
y = -
24
25
× -
15
16
× x
3
×x × z ×z
2
×y = -
24
25
× -
15
16
× x
3+1
× z
1+2
×y
=
9
10
x
4
yz
3
                                                           
Thus, the answer is 
9
10
x
4
yz
3
.
Q u e s t i o n : 1 6
Find each of the following product:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
1
27
a
2
b
2
×
9
2
a
3
b
2
c
2
= -
1
27
×
9
2
× a
2
×a
3
× b
2
×b
2
×c
2
= -
1
27
×
9
2
× a
2+3
× b
2+2
×c
2
= -
1
6
a
5
b
4
c
2
Thus, the answer is -
1
6
a
5
b
4
c
2
.
Q u e s t i o n : 1 7
Find each of the following product:
-7xy ×
1
4
x
2
yz
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-7xy)×
1
4
x
2
yz = -7 ×
1
4
× x ×x
2
×(y ×y)×z = -7 ×
1
4
× x
1+2
× y
1+1
×z = -
7
4
x
3
y
2
z
Thus, the answer is -
7
4
x
3
y
2
z.
Q u e s t i o n : 1 8
Find each of the following product:
(7ab) × (-5ab
2
c) × (6abc
2
)
S o l u t i o n :
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) { ( ) ( )}
( ) ( )
{ ( ) ( )}
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,  a
m
×a
n
= a
m+n
.
We have:
(7ab)× -5ab
2
c × 6abc
2
= {7 ×(-5)×6}×(a ×a ×a)× b ×b
2
×b × c ×c
2
= {7 ×(-5)×6}× a
1+1+1
× b
1+2+1
× c
1+2
= -210a
3
b
4
c
3
Thus, the answer is -210a
3
b
4
c
3
.
Q u e s t i o n : 1 9
Find each of the following product:
(-5a) × (-10a
2
) × (-2a
3
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(-5a)× -10a
2
× -2a
3
= {(-5)×(-10)×(-2)}× a ×a
2
×a
3
= {(-5)×(-10)×(-2)}× a
1+2+3
= -100a
6
Thus, the answer is -100a
6
.
Q u e s t i o n : 2 0
Find each of the following product:
(-4x
2
) × (-6xy
2
) × (-3yz
2
)
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-4x
2
× -6xy
2
× -3yz
2
= {(-4)×(-6)×(-3)}× x
2
×x × y
2
×y ×z
2
= {(-4)×(-6)×(-3)}× x
2+1
× y
2+1
×z
2
= -72x
3
y
3
z
2
Thus, the answer is -72x
3
y
3
z
2
.
Q u e s t i o n : 2 1
Find each of the following product:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
-
2
7
a
4
× -
3
4
a
2
b × -
14
5
b
2
= -
2
7
× -
3
4
× -
14
5
× a
4
×a
2
× b ×b
2
= -
2
7
×
3
4
×
14
5
×a
4+2
×b
1+2
= -
2
7
×
3
4 2
×
14
2 1
5
×a
6
×b
3
= -
3
5
a
6
b
3
Thus, the answer is -
3
5
a
6
b
3
.
Q u e s t i o n : 2 2
Find each of the following product:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
7
9
ab
2
×
15
7
ac
2
b × -
3
5
a
2
c =
7
9
×
15
7
× -
3
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3
7
× -
3
1
5
× a ×a ×a
2
× b
2
×b × c
2
×c =
7
1
9 3
×
15
3 1
7
× -
3
1
5
× a
1+1+2
× b
2+1
× c
2+1
= -a
4
b
3
c
3
Thus, the answer is -a
4
b
3
c
3
.
Q u e s t i o n : 2 3
Find each of the following product:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e.,a
m
×a
n
= a
m+n
.
We have:
4
3
u
2
vw × -5uvw
2
×
1
3
v
2
wu =
4
3
×(-5)×
1
3
× u
2
×u ×u × v ×v ×v
2
× w ×w
2
×w =
4
3
×(-5)×
1
3
× u
2+1+1
× v
1+1+2
× w
1+2+1
= -
20
9
u
4
v
4
w
4
Thus, the answer is -
20
9
u
4
v
4
w
4
.
Q u e s t i o n : 2 4
Find each of the following product:
(0. 5x)×
1
3
xy
2
z
4
× 24x
2
yz
S o l u t i o n :
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( )
( )
( )
( )
( ) { }
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{ }
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( )
( )
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(0. 5x)×
1
3
xy
2
z
4
× 24x
2
yz = 0. 5 ×
1
3
×24 × x ×x ×x
2
× y
2
×y × z
4
×z = 0. 5 ×
1
3
×24 × x
1+1+2
× y
2+1
× z
4+1
= 4x
4
y
3
z
5
Thus, the answer is 4x
4
y
3
z
5
.
Q u e s t i o n : 2 5
Find each of the following product:
4
3
pq
2
× -
1
4
p
2
r × 16p
2
q
2
r
2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
4
3
pq
2
× -
1
4
p
2
r × 16p
2
q
2
r
2
=
4
3
× -
1
4
×16 × p ×p
2
×p
2
× q
2
×q
2
× r×r
2
=
4
3
× -
1
4
×16 × p
1+2+2
× q
2+2
× r
1+2
= -
16
3
p
5
q
4
r
3
Thus, the answer is -
1
3
p
5
q
4
r
3
.
Q u e s t i o n : 2 6
Find each of the following product:
(2.3xy) × (0.1x) × 0.16
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., a
m
×a
n
= a
m+n
.
We have:
(2. 3xy)×(0. 1x)×(0. 16) = (2. 3 ×0. 1 ×0. 16)×(x ×x)×y = (2. 3 ×0. 1 ×0. 16)× x
1+1
×y = 0. 0368x
2
y
Thus, the answer is 0. 0368x
2
y.
Q u e s t i o n : 2 7
Express each of the following product as a monomials and verify the result in each case for x = 1:
(3x) × (4x) × (-5x)
S o l u t i o n :
We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., ? a
m
×a
n
= a
m+n
.
We have:
(3x)×(4x)×(-5x) = {3 ×4 ×(-5)}×(x ×x ×x) = {3 ×4 ×(-5)}× x
1+1+1
= -60x
3
Substituting x = 1 in LHS, we get:
LHS = (3x)×(4x)×(-5x) = (3 ×1)×(4 ×1)×(-5 ×1) = -60
Putting x = 1 in RHS, we get:
RHS = -60x
3
= -60(1)
3
= -60 ×1 = -60
? LHS = RHS for x = 1; therefore, the result is correct
Thus, the answer is -60x
3
.
Q u e s t i o n : 2 8
Express each of the following product as a monomials and verify the result in each case for x = 1:
(4x
2
) × (-3x) × 
4
5
x
3
S o l u t i o n :
We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., ? a
m
×a
n
= a
m+n
.
We  have:
4x
2
×(-3x)×
4
5
x
3
= 4 ×(-3)×
4
5
× x
2
×x ×x
3
= 4 ×(-3)×
4
5
× x
2+1+3
= -
48
5
x
6
? 4x
2
×(-3x)×
4
5
x
3
= -
48
5
x
6
Substituting x = 1 in LHS, we get:
LHS = 4x
2
×(-3x)×
4
5
x
3
= 4 ×1
2
×(-3 ×1)×
4
5
×1
3
= 4 ×(-3)×
4
5
= -
48
5
Putting x = 1 in RHS, we get:
RHS = -
48
5
x
6
= -
48
5
×1
6
= -
48
5
? LHS = RHS for x = 1; therefore, the result is correct
Thus, the answer is -
48
5
x
6
.
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( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
{ ( ) }
( ) ( ) ( )
{ ( ) }
( ) ( ) ( )
( )
( )
( )
( )
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{ }
( )
( )
( )
( )
( )
( )
( )
Q u e s t i o n : 2 9
Express each of the following product as a monomials and verify the result in each case for x = 1:
(5x
4
) × (x
2
)
3
 × (2x)
2
S o l u t i o n :
We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., a
m
×a
n
= a
m+n
 and a
m
n
= a
mn
 .
We have:
5x
4
× x
2
3
×(2x)
2 
= 5x
4
× x
6
× 2
2
×x
2
= 5 ×2
2
× x
4
×x
6
×x
2
= 5 ×2
2
× x
4+6+2
= 20x
12
? 5x
4
× x
2
3
×(2x)
2 
= 20x
12
Substituting x = 1 in LHS, we get:
LHS = 5x
4
× x
2
3
×(2x)
2 
= 5 ×1
4
× 1
2
3
×(2 ×1)
2 
= (5 ×1)× 1
6
×(2)
2 
= 5 ×1 ×4 = 20
Put x =1 in RHS, we get:
RHS = 20x
12
= 20 ×(1)
12
= 20 ×1 = 20
? LHS = RHS for x = 1; therefore, the result is correct.
Thus, the answer is 20x
12
.
Q u e s t i o n : 3 0
Express each of the following product as a monomials and verify the result in each case for x = 1:
(x
2
)
3
 × (2x) × (-4x) × 5
S o l u t i o n :
 We have to find the product of the expression in order to express it as a monomial.
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ?a
m
×a
n
= a
m+n
 and a
m
n
= a
mn
 .
We have:
x
2
3
×(2x)×(-4x)×5 = x
6
×(2x)×(-4x)×5 = {2 ×(-4)×5}× x
6
×x ×x = {2 ×(-4)×5}× x
6+1+1
= -40x
8
? x
2
3
×(2x)×(-4x)×5 = -40x
8
Substituting x = 1 in LHS, we get: ? LHS = x
2
3
×(2x)×(-4x)×5 = 1
2
3
×(2 ×1)×(-4 ×1)×5 = 1
6
×2 ×(-4)×5 = 1 ×2 ×(-4)×5 = -40
Putting x = 1 in RHS, we get: ? RHS = -40x
8
= -40(1)
8
= -40 ×1 = -40
? LHS = RHS for x = 1; therefore, the result is correct
Thus, the answer is -40x
8
.
Q u e s t i o n : 3 1
Write down the product of -8x
2
y
6
 and -20xy. Verify the product for x = 2.5, y = 1.
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ? a
m
×a
n
= a
m+n
.
We have:
-8x
2
y
6
×(-20xy) = {(-8)×(-20)}× x
2
×x × y
6
×y = {(-8)×(-20)}× x
2+1
× y
6+1
= -160x
3
y
7
? -8x
2
y
6
×(-20xy) = -160x
3
y
7
Substituting x = 2.5 and y = 1 in LHS, we get:
LHS = -8x
2
y
6
×(-20xy) = -8(2. 5)
2
(1)
6
×{-20(2. 5)(1)} = {-8(6. 25)(1)}×{-20(2. 5)(1)} = (-50)×(-50) = 2500
Substituting x = 2.5 and y = 1 in RHS, we get: ? RHS = -160x
3
y
7
= -160(2. 5)
3
(1)
7
= -160(15. 625)×1 = -2500
Because LHS is equal to RHS, the result is correct.
Thus, the answer is -160x
3
y
7
.
Q u e s t i o n : 3 2
Evaluate (3.2x
6
y
3
) × (2.1x
2
y
2
) when x = 1 and y = 0.5
Ans
First multiply the expressions and then substitute the values for the variables.
To multiply algebric experssions use the commutative and the associative laws along with the law of indices, a
m
×a
n
= a
m+n
.
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