NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5

Exercise 9.5

Question 1

Use a suitable identity to get each of the following products.

(i) (x + 3) (x + 3) (ii) (2y + 5) (2y + 5)

(iii) (2a ­− 7) (2a − 7) (iv)

(v) (1.1m − 0.4) (1.1 m + 0.4) (vi) (a2 + b2) (− a2 + b2)

(vii) (6x − 7) (6x + 7) (viii) (− a + c) (− a + c)

(ix) (x) (7a − 9b) (7a − 9b)

 

Sol :

The products will be as follows.

(i) (x + 3) (x + 3) = (x + 3)2

= (x)2 + 2(x) (3) + (3)2 [(a + b)2 = a2 + 2ab + b2]

= x2 + 6x + 9

(ii) (2y + 5) (2y + 5) = (2y + 5)2

= (2y)2 + 2(2y) (5) + (5)2 [(a + b)2 = a2 + 2ab + b2]

= 4y2 + 20y + 25

(iii) (2a ­− 7) (2a − 7) = (2a − 7)2

= (2a)2 − 2(2a) (7) + (7)2 [(ab)2 = a2 − 2ab + b2]

= 4a2 − 28a + 49

(iv)

[(ab)2 = a2 − 2ab + b2]

(v) (1.1m − 0.4) (1.1 m + 0.4)

= (1.1m)2 − (0.4)2 [(a + b) (ab) = a2b2]

= 1.21m2 − 0.16

(vi) (a2 + b2) (− a2 + b2) = (b2 + a2) (b2a2)

= (b2)2 − (a2)2 [(a + b) (ab) = a2b2]

= b4a4

(vii) (6x − 7) (6x + 7) = (6x)2 − (7)2 [(a + b) (ab) = a2b2]

= 36x2 − 49

(viii) (− a + c) (− a + c) = (− a + c)2

= (− a)2 + 2(− a) (c) + (c)2 [(a + b)2 = a2 + 2ab + b2]

= a2 − 2ac + c2

(ix)

[(a + b)2 = a2 + 2ab + b2]

(x) (7a − 9b) (7a − 9b) = (7a − 9b)2

= (7a)2 − 2(7a)(9b) + (9b)2 [(a b)2 = a2 − 2ab + b2]

= 49a2 − 126ab + 81b2

 

Question 2

Use the identity (x + a) (x + b) = x2 + (a + b)x + ab to find the following products.

(i) (x + 3) (x + 7) (ii) (4x +5) (4x + 1)

(iii) (4x − 5) (4x − 1) (iv) (4x + 5) (4x − 1)

(v) (2x +5y) (2x + 3y) (vi) (2a2 +9) (2a2 + 5)

(vii) (xyz − 4) (xyz − 2)

 

Sol :

The products will be as follows.

(i) (x + 3) (x + 7) = x2 + (3 + 7) x + (3) (7)

= x2 + 10x + 21

(ii) (4x + 5) (4x + 1) = (4x)2 + (5 + 1) (4x) + (5) (1)

= 16x2 + 24x + 5

(iii)

= 16x2 − 24x + 5

(iv)

= 16x2 + 16x − 5

(v) (2x +5y) (2x + 3y) = (2x)2 + (5y + 3y) (2x) + (5y) (3y)

= 4x2 + 16xy + 15y2

(vi) (2a2 +9) (2a2 + 5) = (2a2)2 + (9 + 5) (2a2) + (9) (5)

= 4a4 + 28a2 + 45

(vii) (xyz − 4) (xyz − 2)

=

= x2y2z2 − 6xyz + 8

 

Question 3

Find the following squares by suing the identities.

(i) (b − 7)2 (ii) (xy + 3z)2 (iii) (6x2 − 5y)2

(iv) (v) (0.4p − 0.5q)2 (vi) (2xy + 5y)2

 

Sol :

(i) (b − 7)2 = (b)2 − 2(b) (7) + (7)2 [(ab)2 = a2 − 2ab + b2]

= b2 − 14b + 49

(ii) (xy + 3z)2 = (xy)2 + 2(xy) (3z) + (3z)2 [(a + b)2 = a2 + 2ab + b2]

= x2y2 + 6xyz + 9z2

(iii) (6x2 − 5y)2 = (6x2)2 − 2(6x2) (5y) + (5y)2 [(ab)2 = a2 − 2ab + b2]

= 36x4 − 60x2y + 25y2

(iv) [(a + b)2 = a2 + 2ab + b2]

(v) (0.4p − 0.5q)2 = (0.4p)2 − 2 (0.4p) (0.5q) + (0.5q)2

[(ab)2 = a2 − 2ab + b2]

= 0.16p2 − 0.4pq + 0.25q2

(vi) (2xy + 5y)2 = (2xy)2 + 2(2xy) (5y) + (5y)2

[(a + b)2 = a2 + 2ab + b2]

= 4x2y2 + 20xy2 + 25y2

 

Question 4

Simplify.

(i) (a2b2)2 (ii) (2x +5)2 − (2x − 5)2

(iii) (7m − 8n)2 + (7m + 8n)2 (iv) (4m + 5n)2 + (5m + 4n)2

(v) (2.5p − 1.5q)2 − (1.5p − 2.5q)2

(vi) (ab + bc)2 − 2ab2c (vii) (m2n2m)2 + 2m3n2

 

Sol :

(i) (a2b2)2 = (a2)2 − 2(a2) (b2) + (b2)2 [(ab)2 = a2 − 2ab + b2 ]

= a4 − 2a2b2 + b4

(ii) (2x +5)2 − (2x − 5)2 = (2x)2 + 2(2x) (5) + (5)2 − [(2x)2 − 2(2x) (5) + (5)2]

[(ab)2 = a2 − 2ab + b2]

[(a + b)2 = a2 + 2ab + b2]

= 4x2 + 20x + 25 − [4x2 − 20x + 25]

= 4x2 + 20x + 25 − 4x2 + 20x − 25 = 40x

(iii) (7m − 8n)2 + (7m + 8n)2

= (7m)2 − 2(7m) (8n) + (8n)2 + (7m)2 + 2(7m) (8n) + (8n)2

[(ab)2 = a2 − 2ab + b2 and (a + b)2 = a2 + 2ab + b2]

= 49m2 − 112mn + 64n2 + 49m2 + 112mn + 64n2

= 98m2 + 128n2

(iv) (4m + 5n)2 + (5m + 4n)2

= (4m)2 + 2(4m) (5n) + (5n)2 + (5m)2 + 2(5m) (4n) + (4n)2

[ (a + b)2 = a2 + 2ab + b2]

= 16m2 + 40mn + 25n2 + 25m2 + 40mn + 16n2

= 41m2 + 80mn + 41n2

(v) (2.5p − 1.5q)2 − (1.5p − 2.5q)2

= (2.5p)2 − 2(2.5p) (1.5q) + (1.5q)2 − [(1.5p)2 − 2(1.5p)(2.5q) + (2.5q)2]

[(ab)2 = a2 − 2ab + b2 ]

= 6.25p2 − 7.5pq + 2.25q2 − [2.25p2 − 7.5pq + 6.25q2]

= 6.25p2 − 7.5pq + 2.25q2 − 2.25p2 + 7.5pq − 6.25q2]

= 4p2 − 4q2

(vi) (ab + bc)2 − 2ab2c

= (ab)2 + 2(ab)(bc) + (bc)2 − 2ab2c [(a + b)2 = a2 + 2ab + b2 ]

= a2b2 + 2ab2c + b2c2 − 2ab2c

= a2b2 + b2c2

(vii) (m2n2m)2 + 2m3n2

= (m2)2 − 2(m2) (n2m) + (n2m)2 + 2m3n2 [(ab)2 = a2 − 2ab + b2 ]

= m4 − 2m3n2 + n4m2 + 2m3n2

= m4 + n4m2

 

Question 5

Show that

(i) (3x + 7)2 − 84x = (3x − 7)2 (ii) (9p − 5q)2 + 180pq = (9p + 5q)2

(iii)

(iv) (4pq + 3q)2 − (4pq − 3q)2 = 48pq2

(v) (ab) (a + b) + (bc) (b + c) + (ca) (c + a) = 0

 

Sol :

(i) L.H.S = (3x + 7)2 − 84x

= (3x)2 + 2(3x)(7) + (7)2 − 84x

= 9x2 + 42x + 49 − 84x

= 9x2 − 42x + 49

R.H.S = (3x − 7)2 = (3x)2 − 2(3x)(7) +(7)2

= 9x2 − 42x + 49

L.H.S = R.H.S

(ii) L.H.S = (9p − 5q)2 + 180pq

= (9p)2 − 2(9p)(5q) + (5q)2 − 180pq

= 81p2 − 90pq + 25q2 + 180pq

= 81p2 + 90pq + 25q2

R.H.S = (9p + 5q)2

= (9p)2 + 2(9p)(5q) + (5q)2

= 81p2 + 90pq + 25q2

L.H.S = R.H.S

(iii) L.H.S =

(iv) L.H.S = (4pq + 3q)2 − (4pq − 3q)2

= (4pq)2 + 2(4pq)(3q) + (3q)2 − [(4pq)2 − 2(4pq) (3q) + (3q)2]

= 16p2q2 + 24pq2 + 9q2 − [16p2q2 − 24pq2 + 9q2]

= 16p2q2 + 24pq2 + 9q2 −16p2q2 + 24pq2 − 9q2

= 48pq2 = R.H.S

(v) L.H.S = (ab) (a + b) + (bc) (b + c) + (ca) (c + a)

= (a2b2) + (b2c2) + (c2a2) = 0 = R.H.S.

 

Question 6

(i) 712 (ii) 992 (iii) 1022 (iv) 9982

(v) (5.2)2 (vi) 297 × 303 (vii) 78 × 82

(viii) 8.92 (ix) 1.05 × 9.5

 

Sol :

(i) 712 = (70 + 1)2

= (70)2 + 2(70) (1) + (1)2 [(a + b)2 = a2 + 2ab + b2 ]

= 4900 + 140 + 1 = 5041

(ii) 992 = (100 − 1)2

= (100)2 − 2(100) (1) + (1)2 [(ab)2 = a2 − 2ab + b2 ]

= 10000 − 200 + 1 = 9801

(iii) 1022 = (100 + 2)2

= (100)2 + 2(100)(2) + (2)2 [(a + b)2 = a2 + 2ab + b2 ]

= 10000 + 400 + 4 = 10404

(iv) 9982 = (1000 − 2)2

= (1000)2 − 2(1000)(2) + (2)2 [(ab)2 = a2 − 2ab + b2 ]

= 1000000 − 4000 + 4 = 996004

(v) (5.2)2 = (5.0 + 0.2)2

= (5.0)2 + 2(5.0) (0.2) + (0.2)2 [(a + b)2 = a2 + 2ab + b2 ]

= 25 + 2 + 0.04 = 27.04

(vi) 297 × 303 = (300 − 3) × (300 + 3)

= (300)2 − (3)2 [(a + b) (ab) = a2b2]

= 90000 − 9 = 89991

(vii) 78 × 82 = (80 − 2) (80 + 2)

= (80)2 − (2)2 [(a + b) (ab) = a2b2]

= 6400 − 4 = 6396

(viii) 8.92 = (9.0 − 0.1)2

= (9.0)2 − 2(9.0) (0.1) + (0.1)2 [(ab)2 = a2 − 2ab + b2 ]

= 81 − 1.8 + 0.01 = 79.21

(ix) 1.05 × 9.5 = 1.05 × 0.95 × 10

= (1 + 0.05) (1− 0.05) ×10

= [(1)2 − (0.05)2] × 10

= [1 − 0.0025] × 10 [(a + b) (ab) = a2b2]

= 0.9975 × 10 = 9.975

 

Question 7

Using a2 b2 = (a + b) (ab), find

(i) 512 − 492 (ii) (1.02)2 − (0.98)2 (iii) 1532 − 1472

(iv) 12.12 − 7.92

 

So :

(i) 512 − 492 = (51 + 49) (51 − 49)

= (100) (2) = 200

(ii) (1.02)2 − (0.98)2 = (1.02 + 0.98) (1.02 ­− 0.98)

= (2) (0.04) = 0.08

(iii) 1532 − 1472 = (153 + 147) (153 − 147)

= (300) (6) = 1800

(iv) 12.12 − 7.92 = (12.1 + 7.9) (12.1 − 7.9)

= (20.0) (4.2) = 84

 

Question 8

Using (x + a) (x + b) = x2 + (a + b) x + ab, find

(i) 103 × 104 (ii) 5.1 × 5.2 (iii) 103 × 98 (iv) 9.7 × 9.8

 

Sol :

(i) 103 × 104 = (100 + 3) (100 + 4)

= (100)2 + (3 + 4) (100) + (3) (4)

= 10000 + 700 + 12 = 10712

(ii) 5.1 × 5.2 = (5 + 0.1) (5 + 0.2)

= (5)2 + (0.1 + 0.2) (5) + (0.1) (0.2)

= 25 + 1.5 + 0.02 = 26.52

(iii) 103 × 98 = (100 + 3) (100 − 2)

= (100)2 + [3 + (− 2)] (100) + (3) (− 2)

= 10000 + 100 − 6

= 10094

(iv) 9.7 × 9.8 = (10 − 0.3) (10 − 0.2)

= (10)2 + [(− 0.3) + (− 0.2)] (10) + (− 0.3) (− 0.2)

= 100 + (− 0.5)10 + 0.06 = 100.06 − 5 = 95.06

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