RS Aggarwal Class 8 Math Sixth Chapter Operation on Algebric Expressions Exercise 6D Solution

RS Aggarwal Class 8 Math Sixth Chapter Operation on Algebric Expressions Exercise 6D Solution

EXERCISE 6D

(1) Find each of the following products:

(i) (x + 6) (x + 6)

= x2 + 6x + 6x + 36

= x2 + 12x + 36

= x2 + (2 × x × 6) + 62

= (x + 6)2

(ii) (4x + 5y) (4x + 5y)

= 16x2 + 20xy + 20xy + 25y2

= (4x)2 + (2 × 4x × 5y) + (5y)2

= (4x + 5y)2

(iii) (7a + 9b) (7a + 9b)

= 49a2 + 63ab + 63ab + 81b2

= (7a)2 + (2 × 7a × 9b) + (9b)2

= (7a + 9b)2

(v) (x2 + 7) (x2 + 7)

= x4 + 7x2 + 7x2 + 49

= (x2)2 + (2 × x2 × 7) + 72

= (x2 + 7)2

(2) Find each of the following products:

(i) (x – 4) (x – 4)

= x2 – 4x – 4x + 16

= x2 – 8x + 16

= x2 – (2 × x × 4) + 42

= (x – 4)2

(ii) (2x – 3y) (2x – 3y)

= 4x2 – 6xy – 6xy + 9y2

= (2x)2 – 12xy + (3y)2

= (2x)2 – (2 × 2x × 3y) + (3y)2

= (2x – 3y)2




(3) Expand:

(i) (8a + 3b)2

= (8a)2 + 2 × 8a × 3b + (3b)2

= 64a2 + 48ab + 9b2

(ii) (7x + 2y)2

= (7x)2 + (2 × 7x × 2y) + (2y)2

= 49x2 + 28xy + (2y)2

(iii) (5x + 11)2

= (5x)2 + (2 × 5x × 11) + (11)2

= 25x2 + 110x + 121

(vi) (9x – 10)2

= (9x)2 – (2 × 9x × 10) + (10)2

= 81x2 – 180x + 100

(vii) (x2y – yz2)2

= (x2y)2 – (2 × x2y × yz2) + (yz2)2

= x4y2 – 2x2y2z2 + y2z4

(4) Find each of the following products:

(i) (x + 3) (x – 3)

= x2 + 3x – 3x – 9

= x2 – 32

(ii) (2x + 5) (2x – 5)

= 4x2 + 10x – 10x – 25

= (2x)2 – 52

(iii) (8 + x) (8 – x)

= 64 + 8x – 8x – x2

= 42 – x2

(iv) (7x + 11y) (7x – 11y)

= 49x2 + 77xy – 77xy – 121y2

= (7x)2 – (11y)2


(5) Using the formula for squaring a binomial, evaluate the following:

(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

(6) Using the formula for squaring a binomial, evaluate the following:

(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) + (2)2

= 6400 – 320 + 4

= 6084

(iii) (197)2

= (200 – 3)2

= (200)2 – (2 × 200 × 3) + (3)2

= 40000 – 1200 + 9

= 38809

(iv) (999)2

= (1000 – 1)2

= (1000)2 – (2 × 1000 × 1) + 1

= 1000000 – 2000 + 1

= 998001

(7) Find the value of:

(i) (82)2 – (18)2

= [(80+2)2] – [(20 – 2)2]

= [(80)2 + (2 × 80 × 2) + 4] – [(20)2 – (2 × 20 × 2) + 4]

= (6400 + 320 + 4) – (400 – 80 + 4)

= 6724 – 324

= 6400

(ii) (128)2 – (72)2

= [(130 – 2)2 – (70 + 2)2]

= [(130)2 – (2 × 130 × 2) + 4] – [(70)2 + (2 × 70 × 2) + 4]

= (16900 – 520 + 4) – (4900 + 280 + 4)

= 16384 – 5184

= 11200

(iii) 197 × 203

= (200 – 3) × (200 + 3)

= (200)2 – (3)2

= 40000 – 9

= 39991

(v) (14.7 × 15.3)

= (15 – 0.3) × (15 + 0.3)

= (15)2 – (0.3)2

= 225 – 0.09

= 224.91

(vi) (8.63)2 – (1.37)2

= (8.63 + 1.37) (8.63 – 1.37)

= 10 × 7.26

= 72.6

(8) Find the value of the expression (9x2 + 24x + 16), when x = 12.

Solution: 9x2 + 24x + 16

= [9 × (12)2] + (24 × 12) + 16

= (9 × 144) + 288 + 16

= 1296 + 288 + 16

= 1600

(9) Find the value of the expression (64x2 + 81y2 + 144xy), when x = 11 and y = 4/3.

Solution:  64x2 + 81y2 + 144xy

(10) Find the value of the expression (36x2 + 25y2 – 60xy), when x =2/3 and y = 1/5.

Solution: (36x2 + 25y2 – 60xy)

= (6x)2 – (2 × 6x × 5y) + (5y)2

= (6x – 5y)2





(13) Find the continued product:

(i) (x + 1) (x – 1) (x2 + 1)

= (x2 – 1) (x2 + 1)

= x4 – 1

(ii) (x – 3) (x + 3) (x2 + 9)

= [(x)2 – (3)2] (x2 + 9)

= (x2 – 9)(x2 + 9)

= x2 – 92 = x2 – 81

(iii) (3x – 2y) (3x + 2y) (9x2 + 4y2)

= [(3x)2 – (2y2)] (9x2 + 4y2)

= (9x2 – 4y2) (9x2 + 4y2)

= (9x2)2 – (4y2)2

= 81x4 – 16y4

(iv) (2p + 3) (2p – 3) (4p2 + 9)

= [(2p)2 – (3)2] (4p2 + 9)

= (4p2 – 9) (4p2 + 9)

= (4p2)2 – (9)2

= 16p4 – 81

(14) If x + y = 12 and xy = 14, find the value of (x2 + y2).

Solution: x + y = 12

⇒ (x + y)2 = (12)2

⇒ x2 + 2xy + y2 = 144

⇒ (x2 + y2) + (2 × 14) = 144

⇒ (x2 + y2) = 144 – 28

⇒ (x2 + y2) = 116

(15) If x – y = 7 and xy = 9, find the value of (x2 + y2).

Solution: x – y = 7

⇒ (x – y)2 = (7)2

⇒ x2 – 2xy + y2 = 49

⇒ (x2 + y2) – (2 × 9) = 49

⇒ (x2 + y2) = 49 – 18

⇒ (x2 + y2) = 31.


3 Comments

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  1. Thank you R S Aggarwal ji

  2. `SuPeR StAr AyUsH,

    thanxxx net explantions

  3. Thank you for helping us

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