Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > RS Aggarwal Solutions: Exercise 6D - Operations on Algebraic Expressions
RS Aggarwal Solutions: Exercise 6D - Operations on Algebraic Expressions | Mathematics (Maths) Class 8 PDF Download
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Page 1
( ) ( ) ( [ ( ]
Page 2
( ) ( ) ( [ ( ]
4x +5y 4x +5y = 4x +5y)
2
= (4x)
2
+(5y)
2
+2 ×4x ×5y using a +b)
2
= a
2
+b
2
+2ab = 16x
2
+25y
2
+40xy
iii
We have:
7a +9b 7a +9b = 7a +9b)
2
= (7a)
2
+(9b)
2
+2 ×7a ×9b using a +b)
2
= a
2
+b
2
+2ab = 49a
2
+81b
2
+126ab
iv
We have:
2
3
x +
4
5
y
2
3
x +
4
5
y =
2
3
x +
4
5
y
2
=
2
3
x
2
+
4
5
y
2
+2 ×
2
3
x ×
4
5
y using a +b)
2
= a
2
+b
2
+2ab =
4
9
x
2
+
16
25
y
2
+
16
15
xy
v
We have:
x
2
+7 x
2
+7 = x
2
+7)
2
= x
2
2
+7
2
+2 ×x
2
×7 using a +b)
2
= a
2
+b
2
+2ab = x
4
+49 +14x
2
vi
We have:
5
6
a
2
+2
5
6
a
2
+2 =
5
6
a
2
+2
2
=
5
6
a
2
2
+(2)
2
+2 ×
5
6
a
2
×2 using a +b)
2
= a
2
+b
2
+2ab =
25
36
a
4
+4 +
10
3
a
2
Q u e s t i o n : 6 3
Find each of the following products:
i
(x - 4)(x - 4)
ii
(2x - 3y)(2x - 3y)
iii
3
4
x -
5
6
y
3
4
x -
5
6
y
iv
x -
3
x
x -
3
x
v
1
3
x
2
-9
1
3
x
2
-9
vi
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y
S o l u t i o n :
i
We have:
x -4 x -4 = x -4)
2
= x
2
-2 ×x ×4 +4
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-8x +16
ii
We have:
2x -3y 2x -3y = 2x -3y)
2
= (2x)
2
-2 ×2x ×3y +(3y)
2
using a -b)
2
= a
2
-2ab +b
2
= 4x
2
-12xy +9y
2
iii
We have:
3
4
x -
5
6
y
3
4
x -
5
6
y =
3
4
x -
5
6
y
2
=
3
4
x
2
-2 ×
3
4
x ×
5
6
y +
5
6
y
2
using a -b)
2
= a
2
-2ab +b
2
=
9
16
x
2
-
15
12
xy +
25
36
y
2
iv
We have:
x -
3
x
x -
3
x
= x -
3
x
2
= (x)
2
-2 ×x ×
3
x
+
3
x
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-6 +
9
x
2
v
We have:
1
3
x
2
-9
1
3
x
2
-9 =
1
3
x
2
-9
2
=
1
3
x
2
2
-2 ×
1
3
x
2
×9 +(9)
2
using a -b)
2
= a
2
-2ab +b
2
=
1
9
x
4
-6x
2
+81
vi
We have:
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y =
1
2
y
2
-
1
3
y
2
=
1
2
y
2
2
-2 ×
1
2
y
2
×
1
3
y +
1
3
y
2
using a -b)
2
= a
2
-2ab +b
2
=
1
4
y
4
-
1
3
y
3
+
1
9
y
2
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( )( ) (
( )
[ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
Page 3
( ) ( ) ( [ ( ]
4x +5y 4x +5y = 4x +5y)
2
= (4x)
2
+(5y)
2
+2 ×4x ×5y using a +b)
2
= a
2
+b
2
+2ab = 16x
2
+25y
2
+40xy
iii
We have:
7a +9b 7a +9b = 7a +9b)
2
= (7a)
2
+(9b)
2
+2 ×7a ×9b using a +b)
2
= a
2
+b
2
+2ab = 49a
2
+81b
2
+126ab
iv
We have:
2
3
x +
4
5
y
2
3
x +
4
5
y =
2
3
x +
4
5
y
2
=
2
3
x
2
+
4
5
y
2
+2 ×
2
3
x ×
4
5
y using a +b)
2
= a
2
+b
2
+2ab =
4
9
x
2
+
16
25
y
2
+
16
15
xy
v
We have:
x
2
+7 x
2
+7 = x
2
+7)
2
= x
2
2
+7
2
+2 ×x
2
×7 using a +b)
2
= a
2
+b
2
+2ab = x
4
+49 +14x
2
vi
We have:
5
6
a
2
+2
5
6
a
2
+2 =
5
6
a
2
+2
2
=
5
6
a
2
2
+(2)
2
+2 ×
5
6
a
2
×2 using a +b)
2
= a
2
+b
2
+2ab =
25
36
a
4
+4 +
10
3
a
2
Q u e s t i o n : 6 3
Find each of the following products:
i
(x - 4)(x - 4)
ii
(2x - 3y)(2x - 3y)
iii
3
4
x -
5
6
y
3
4
x -
5
6
y
iv
x -
3
x
x -
3
x
v
1
3
x
2
-9
1
3
x
2
-9
vi
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y
S o l u t i o n :
i
We have:
x -4 x -4 = x -4)
2
= x
2
-2 ×x ×4 +4
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-8x +16
ii
We have:
2x -3y 2x -3y = 2x -3y)
2
= (2x)
2
-2 ×2x ×3y +(3y)
2
using a -b)
2
= a
2
-2ab +b
2
= 4x
2
-12xy +9y
2
iii
We have:
3
4
x -
5
6
y
3
4
x -
5
6
y =
3
4
x -
5
6
y
2
=
3
4
x
2
-2 ×
3
4
x ×
5
6
y +
5
6
y
2
using a -b)
2
= a
2
-2ab +b
2
=
9
16
x
2
-
15
12
xy +
25
36
y
2
iv
We have:
x -
3
x
x -
3
x
= x -
3
x
2
= (x)
2
-2 ×x ×
3
x
+
3
x
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-6 +
9
x
2
v
We have:
1
3
x
2
-9
1
3
x
2
-9 =
1
3
x
2
-9
2
=
1
3
x
2
2
-2 ×
1
3
x
2
×9 +(9)
2
using a -b)
2
= a
2
-2ab +b
2
=
1
9
x
4
-6x
2
+81
vi
We have:
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y =
1
2
y
2
-
1
3
y
2
=
1
2
y
2
2
-2 ×
1
2
y
2
×
1
3
y +
1
3
y
2
using a -b)
2
= a
2
-2ab +b
2
=
1
4
y
4
-
1
3
y
3
+
1
9
y
2
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( )( ) (
( )
[ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
Q u e s t i o n : 6 4
Expand:
i
(8a + 3b)
2
ii
(7x + 2y)
2
iii
(5x + 11)
2
iv
a
2
+
2
a
2
v
3x
4
+
2y
9
2
vi
(9x - 10)
2
vii
(x
2
y - yz
2
)
2
viii
x
y
-
y
x
2
ix
3m -
4
5
n
2
S o l u t i o n :
We shall use the identities (a+b)
2
=a
2
+b
2
+2ab and (a-b)
2
=a
2
+b
2
-2ab.
i
We have:
8a +3b)
2
= (8a)
2
+2 ×8a ×3b +(3b)
2
= 64a
2
+48ab +9b
2
ii
We have:
7x +2y)
2
= (7x)
2
+2 ×7x ×2y +(2y)
2
= 49x
2
+28xy +4y
2
iii
We have :
5x +11)
2
= (5x)
2
+2 ×5x ×11 +(11)
2
= 25x
2
+110x +121
iv
We have:
a
2
+
2
a
2
=
a
2
2
+2 ×
a
2
×
2
a
+
2
a
2
=
a
4
2
+2 +
4
a
2
v
We have:
3x
4
+
2y
9
2
=
3x
4
2
+2 ×
3x
4
×
2y
9
+
2y
9
2
=
9x
16
2
+
1
3
xy +
4y
2
81
vi
We have:
9x -10)
2
(9x)
2
-2 ×9x ×10 +(10)
2
= 81x
2
-180x +100
vii
We have:
x
2
y -yz
2
)
2
x
2
y
2
-2 ×x
2
y ×yz
2
+ yz
2
2
= x
4
y
2
-2x
2
y
2
z
2
+y
2
z
4
viii
We have:
x
y
-
y
x
2
=
x
y
2
-2 ×
x
y
×
y
x
+
y
x
2
=
x
2
y
2
-2 +
y
2
x
2
ix
We have:
3m -
4
5
n
2
= (3m)
2
-2 ×3m ×
4
5
n +
4
5
n
2
= 9m
2
-
24mn
5
+
16
25
n
2
Q u e s t i o n : 6 5
Find each of the following products:
i
(x + 3)(x - 3)
( )
( )
( )
( )
(
(
(
( ) ( ) ( )
( ) ( ) ( )
(
(
( ) ( )
( ) ( ) ( )
( ) ( )
Page 4
( ) ( ) ( [ ( ]
4x +5y 4x +5y = 4x +5y)
2
= (4x)
2
+(5y)
2
+2 ×4x ×5y using a +b)
2
= a
2
+b
2
+2ab = 16x
2
+25y
2
+40xy
iii
We have:
7a +9b 7a +9b = 7a +9b)
2
= (7a)
2
+(9b)
2
+2 ×7a ×9b using a +b)
2
= a
2
+b
2
+2ab = 49a
2
+81b
2
+126ab
iv
We have:
2
3
x +
4
5
y
2
3
x +
4
5
y =
2
3
x +
4
5
y
2
=
2
3
x
2
+
4
5
y
2
+2 ×
2
3
x ×
4
5
y using a +b)
2
= a
2
+b
2
+2ab =
4
9
x
2
+
16
25
y
2
+
16
15
xy
v
We have:
x
2
+7 x
2
+7 = x
2
+7)
2
= x
2
2
+7
2
+2 ×x
2
×7 using a +b)
2
= a
2
+b
2
+2ab = x
4
+49 +14x
2
vi
We have:
5
6
a
2
+2
5
6
a
2
+2 =
5
6
a
2
+2
2
=
5
6
a
2
2
+(2)
2
+2 ×
5
6
a
2
×2 using a +b)
2
= a
2
+b
2
+2ab =
25
36
a
4
+4 +
10
3
a
2
Q u e s t i o n : 6 3
Find each of the following products:
i
(x - 4)(x - 4)
ii
(2x - 3y)(2x - 3y)
iii
3
4
x -
5
6
y
3
4
x -
5
6
y
iv
x -
3
x
x -
3
x
v
1
3
x
2
-9
1
3
x
2
-9
vi
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y
S o l u t i o n :
i
We have:
x -4 x -4 = x -4)
2
= x
2
-2 ×x ×4 +4
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-8x +16
ii
We have:
2x -3y 2x -3y = 2x -3y)
2
= (2x)
2
-2 ×2x ×3y +(3y)
2
using a -b)
2
= a
2
-2ab +b
2
= 4x
2
-12xy +9y
2
iii
We have:
3
4
x -
5
6
y
3
4
x -
5
6
y =
3
4
x -
5
6
y
2
=
3
4
x
2
-2 ×
3
4
x ×
5
6
y +
5
6
y
2
using a -b)
2
= a
2
-2ab +b
2
=
9
16
x
2
-
15
12
xy +
25
36
y
2
iv
We have:
x -
3
x
x -
3
x
= x -
3
x
2
= (x)
2
-2 ×x ×
3
x
+
3
x
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-6 +
9
x
2
v
We have:
1
3
x
2
-9
1
3
x
2
-9 =
1
3
x
2
-9
2
=
1
3
x
2
2
-2 ×
1
3
x
2
×9 +(9)
2
using a -b)
2
= a
2
-2ab +b
2
=
1
9
x
4
-6x
2
+81
vi
We have:
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y =
1
2
y
2
-
1
3
y
2
=
1
2
y
2
2
-2 ×
1
2
y
2
×
1
3
y +
1
3
y
2
using a -b)
2
= a
2
-2ab +b
2
=
1
4
y
4
-
1
3
y
3
+
1
9
y
2
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( )( ) (
( )
[ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
Q u e s t i o n : 6 4
Expand:
i
(8a + 3b)
2
ii
(7x + 2y)
2
iii
(5x + 11)
2
iv
a
2
+
2
a
2
v
3x
4
+
2y
9
2
vi
(9x - 10)
2
vii
(x
2
y - yz
2
)
2
viii
x
y
-
y
x
2
ix
3m -
4
5
n
2
S o l u t i o n :
We shall use the identities (a+b)
2
=a
2
+b
2
+2ab and (a-b)
2
=a
2
+b
2
-2ab.
i
We have:
8a +3b)
2
= (8a)
2
+2 ×8a ×3b +(3b)
2
= 64a
2
+48ab +9b
2
ii
We have:
7x +2y)
2
= (7x)
2
+2 ×7x ×2y +(2y)
2
= 49x
2
+28xy +4y
2
iii
We have :
5x +11)
2
= (5x)
2
+2 ×5x ×11 +(11)
2
= 25x
2
+110x +121
iv
We have:
a
2
+
2
a
2
=
a
2
2
+2 ×
a
2
×
2
a
+
2
a
2
=
a
4
2
+2 +
4
a
2
v
We have:
3x
4
+
2y
9
2
=
3x
4
2
+2 ×
3x
4
×
2y
9
+
2y
9
2
=
9x
16
2
+
1
3
xy +
4y
2
81
vi
We have:
9x -10)
2
(9x)
2
-2 ×9x ×10 +(10)
2
= 81x
2
-180x +100
vii
We have:
x
2
y -yz
2
)
2
x
2
y
2
-2 ×x
2
y ×yz
2
+ yz
2
2
= x
4
y
2
-2x
2
y
2
z
2
+y
2
z
4
viii
We have:
x
y
-
y
x
2
=
x
y
2
-2 ×
x
y
×
y
x
+
y
x
2
=
x
2
y
2
-2 +
y
2
x
2
ix
We have:
3m -
4
5
n
2
= (3m)
2
-2 ×3m ×
4
5
n +
4
5
n
2
= 9m
2
-
24mn
5
+
16
25
n
2
Q u e s t i o n : 6 5
Find each of the following products:
i
(x + 3)(x - 3)
( )
( )
( )
( )
(
(
(
( ) ( ) ( )
( ) ( ) ( )
(
(
( ) ( )
( ) ( ) ( )
( ) ( )
ii
(2x + 5)(2x - 5)
iii
(8 + x)(8 - x)
iv
(7x + 11y)(7x - 11y)
v
5x
2
+
3
4
y
2
5x
2
-
3
4
y
2
vi
4x
5
-
5y
3
4x
5
+
5y
3
vii
x +
1
x
x -
1
x
viii
1
x
+
1
y
1
x
-
1
y
S o l u t i o n :
i
We have:
x +3 x -3 = x
2
-9 using a +b a -b = a
2
-b
2
ii
We have:
2x +5 2x -5 = 4x
2
-25 using a +b a -b = a
2
-b
2
iii
We have:
8 +x 8 -x = 64 -x
2
using a +b a -b = a
2
-b
2
iv
We have:
7x +11y 7x -11y = 49x
2
-121y
2
using a +b a -b = a
2
-b
2
v
We have:
5x
2
+
3
4
y
2
5x
2
-
3
4
y
2
= 25x
4
-
9
16
y
4
using a +b a -b = a
2
-b
2
vi
We have:
4x
5
-
5y
3
4x
5
+
5y
3
=
16x
2
25
-
25y
2
9
using a +b a -b = a
2
-b
2
vii
We have:
x +
1
x
x -
1
x
= x
2
-
1
x
2
using a +b a -b = a
2
-b
2
viii
We have:
1
x
+
1
y
1
x
-
1
y
=
1
x
2
-
1
y
2
using a +b a -b = a
2
-b
2
ix
We have:
2a +
3
b
2a -
3
b
= 4a
2
-
9
b
2
using a +b a -b = a
2
-b
2
Q u e s t i o n : 6 6
Using the formula for squaring a binomial, evaluate the following:
i
54
2
ii
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( )( ) )]
( ) ( ) [ ( )( ) ]
( ) ( ) [ ( )( ) ]
( ) ( ) [ ( )( ) ]
Page 5
( ) ( ) ( [ ( ]
4x +5y 4x +5y = 4x +5y)
2
= (4x)
2
+(5y)
2
+2 ×4x ×5y using a +b)
2
= a
2
+b
2
+2ab = 16x
2
+25y
2
+40xy
iii
We have:
7a +9b 7a +9b = 7a +9b)
2
= (7a)
2
+(9b)
2
+2 ×7a ×9b using a +b)
2
= a
2
+b
2
+2ab = 49a
2
+81b
2
+126ab
iv
We have:
2
3
x +
4
5
y
2
3
x +
4
5
y =
2
3
x +
4
5
y
2
=
2
3
x
2
+
4
5
y
2
+2 ×
2
3
x ×
4
5
y using a +b)
2
= a
2
+b
2
+2ab =
4
9
x
2
+
16
25
y
2
+
16
15
xy
v
We have:
x
2
+7 x
2
+7 = x
2
+7)
2
= x
2
2
+7
2
+2 ×x
2
×7 using a +b)
2
= a
2
+b
2
+2ab = x
4
+49 +14x
2
vi
We have:
5
6
a
2
+2
5
6
a
2
+2 =
5
6
a
2
+2
2
=
5
6
a
2
2
+(2)
2
+2 ×
5
6
a
2
×2 using a +b)
2
= a
2
+b
2
+2ab =
25
36
a
4
+4 +
10
3
a
2
Q u e s t i o n : 6 3
Find each of the following products:
i
(x - 4)(x - 4)
ii
(2x - 3y)(2x - 3y)
iii
3
4
x -
5
6
y
3
4
x -
5
6
y
iv
x -
3
x
x -
3
x
v
1
3
x
2
-9
1
3
x
2
-9
vi
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y
S o l u t i o n :
i
We have:
x -4 x -4 = x -4)
2
= x
2
-2 ×x ×4 +4
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-8x +16
ii
We have:
2x -3y 2x -3y = 2x -3y)
2
= (2x)
2
-2 ×2x ×3y +(3y)
2
using a -b)
2
= a
2
-2ab +b
2
= 4x
2
-12xy +9y
2
iii
We have:
3
4
x -
5
6
y
3
4
x -
5
6
y =
3
4
x -
5
6
y
2
=
3
4
x
2
-2 ×
3
4
x ×
5
6
y +
5
6
y
2
using a -b)
2
= a
2
-2ab +b
2
=
9
16
x
2
-
15
12
xy +
25
36
y
2
iv
We have:
x -
3
x
x -
3
x
= x -
3
x
2
= (x)
2
-2 ×x ×
3
x
+
3
x
2
using a -b)
2
= a
2
-2ab +b
2
= x
2
-6 +
9
x
2
v
We have:
1
3
x
2
-9
1
3
x
2
-9 =
1
3
x
2
-9
2
=
1
3
x
2
2
-2 ×
1
3
x
2
×9 +(9)
2
using a -b)
2
= a
2
-2ab +b
2
=
1
9
x
4
-6x
2
+81
vi
We have:
1
2
y
2
-
1
3
y
1
2
y
2
-
1
3
y =
1
2
y
2
-
1
3
y
2
=
1
2
y
2
2
-2 ×
1
2
y
2
×
1
3
y +
1
3
y
2
using a -b)
2
= a
2
-2ab +b
2
=
1
4
y
4
-
1
3
y
3
+
1
9
y
2
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( )( ) (
( )
[ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( [ ( ]
( ) ( ) ( [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) [ ( ]
( ) ( ) ( ) ( ) ( ) [ ( ]
Q u e s t i o n : 6 4
Expand:
i
(8a + 3b)
2
ii
(7x + 2y)
2
iii
(5x + 11)
2
iv
a
2
+
2
a
2
v
3x
4
+
2y
9
2
vi
(9x - 10)
2
vii
(x
2
y - yz
2
)
2
viii
x
y
-
y
x
2
ix
3m -
4
5
n
2
S o l u t i o n :
We shall use the identities (a+b)
2
=a
2
+b
2
+2ab and (a-b)
2
=a
2
+b
2
-2ab.
i
We have:
8a +3b)
2
= (8a)
2
+2 ×8a ×3b +(3b)
2
= 64a
2
+48ab +9b
2
ii
We have:
7x +2y)
2
= (7x)
2
+2 ×7x ×2y +(2y)
2
= 49x
2
+28xy +4y
2
iii
We have :
5x +11)
2
= (5x)
2
+2 ×5x ×11 +(11)
2
= 25x
2
+110x +121
iv
We have:
a
2
+
2
a
2
=
a
2
2
+2 ×
a
2
×
2
a
+
2
a
2
=
a
4
2
+2 +
4
a
2
v
We have:
3x
4
+
2y
9
2
=
3x
4
2
+2 ×
3x
4
×
2y
9
+
2y
9
2
=
9x
16
2
+
1
3
xy +
4y
2
81
vi
We have:
9x -10)
2
(9x)
2
-2 ×9x ×10 +(10)
2
= 81x
2
-180x +100
vii
We have:
x
2
y -yz
2
)
2
x
2
y
2
-2 ×x
2
y ×yz
2
+ yz
2
2
= x
4
y
2
-2x
2
y
2
z
2
+y
2
z
4
viii
We have:
x
y
-
y
x
2
=
x
y
2
-2 ×
x
y
×
y
x
+
y
x
2
=
x
2
y
2
-2 +
y
2
x
2
ix
We have:
3m -
4
5
n
2
= (3m)
2
-2 ×3m ×
4
5
n +
4
5
n
2
= 9m
2
-
24mn
5
+
16
25
n
2
Q u e s t i o n : 6 5
Find each of the following products:
i
(x + 3)(x - 3)
( )
( )
( )
( )
(
(
(
( ) ( ) ( )
( ) ( ) ( )
(
(
( ) ( )
( ) ( ) ( )
( ) ( )
ii
(2x + 5)(2x - 5)
iii
(8 + x)(8 - x)
iv
(7x + 11y)(7x - 11y)
v
5x
2
+
3
4
y
2
5x
2
-
3
4
y
2
vi
4x
5
-
5y
3
4x
5
+
5y
3
vii
x +
1
x
x -
1
x
viii
1
x
+
1
y
1
x
-
1
y
S o l u t i o n :
i
We have:
x +3 x -3 = x
2
-9 using a +b a -b = a
2
-b
2
ii
We have:
2x +5 2x -5 = 4x
2
-25 using a +b a -b = a
2
-b
2
iii
We have:
8 +x 8 -x = 64 -x
2
using a +b a -b = a
2
-b
2
iv
We have:
7x +11y 7x -11y = 49x
2
-121y
2
using a +b a -b = a
2
-b
2
v
We have:
5x
2
+
3
4
y
2
5x
2
-
3
4
y
2
= 25x
4
-
9
16
y
4
using a +b a -b = a
2
-b
2
vi
We have:
4x
5
-
5y
3
4x
5
+
5y
3
=
16x
2
25
-
25y
2
9
using a +b a -b = a
2
-b
2
vii
We have:
x +
1
x
x -
1
x
= x
2
-
1
x
2
using a +b a -b = a
2
-b
2
viii
We have:
1
x
+
1
y
1
x
-
1
y
=
1
x
2
-
1
y
2
using a +b a -b = a
2
-b
2
ix
We have:
2a +
3
b
2a -
3
b
= 4a
2
-
9
b
2
using a +b a -b = a
2
-b
2
Q u e s t i o n : 6 6
Using the formula for squaring a binomial, evaluate the following:
i
54
2
ii
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( ) ( ) ]
( ) ( ) [ ( )( ) )]
( ) ( ) [ ( )( ) ]
( ) ( ) [ ( )( ) ]
( ) ( ) [ ( )( ) ]
82
2
iii
103
2
iv
704
2
S o l u t i o n :
We shall use the identity (a+b)
2
=a
2
+b
2
+2ab.
i
(54)
2
= 50 +4)
2
= (50)
2
+2 ×50 ×4 +(4)
2
= 2500 +400 +16 = 2916
ii
(82)
2
= 80 +2)
2
= (80)
2
+2 ×80 ×2 +(2)
2
= 6400 +320 +4 = 6724
iii
(103)
2
= 100 +3)
2
= (100)
2
+2 ×100 ×3 +(3)
2
= 10000 +600 +9 = 10609
iv
(704)
2
= 700 +4)
2
= (700)
2
+2 ×700 ×4 +(4)
2
= 490000 +5600 +16 = 495616
Q u e s t i o n : 6 7
Using the formula for squaring a binomial, evaluate the following:
i
69
2
ii
78
2
iii
197
2
iv
999
2
S o l u t i o n :
We shall use the identity (a-b)
2
= a
2
+b
2
-2ab.
i
(69)
2
= 70 -1)
2
= (70)
2
-2 ×70 ×1 +1 = 4900 -140 +1 = 4761
ii
(78)
2
= 80 -2)
2
= (80)
2
-2 ×80 ×2 +4 = 6400 -320 +4 = 6084
iii
(197)
2
= 200 -3)
2
= (200)
2
-2 ×200 ×3 +9 = 40000 -1200 +9 = 38809
iv
(999)
2
= 1000 -1)
2
= (1000)
2
-2 ×1000 ×1 +1 = 1000000 -2000 +1 = 998001
Q u e s t i o n : 6 8
Find the value of:
i
82
2
- 18
2
ii
128
2
- 72
2
iii
197 × 203
iv
198×198-102×102
96
v
14.7 ×15.3
(
(
(
(
(
(
(
(
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Document Description: RS Aggarwal Solutions: Exercise 6D - Operations on Algebraic Expressions for Class 8 2024 is part of Mathematics (Maths) Class 8 preparation.
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