Exercise 11.4

1. Select a suitable identity and find the following products

(i) (3k + 4l) (3k + 4l)

Solution:

Using a+b2 = a2a3 + 2abab + b2b, where a = and b = :

3k+4l2 = (3k23k) + 2()() + (4l24l)

= 9k2−9k2 + kl + 16l24l2

(ii) ax2+by2ax2+by2

Solution:

Using a+b2 = a2a3 + 2abab + b2b, where a = ax2ax and b = by2by:

ax2+by22 = ax22 + 2ax2by2 + by22

= a2x4a2x2 + 2abx2y22abx + b2y4b2y2

(iii) (7d – 9e) (7d – 9e)

Solution:

Using a−b2 = a2a3 - 2abab + b2b, where a = and b = :

7d−9e2 = 7d2 - 2(7d)(9e) + 9e2

= 49d249d - de + 81e29e2

(iv) m2−n2m2+n2

Solution:

Using (a - b)(a + b) = a22a - b22b, where a = m2m3 and b = n2n3:

m2−n2m2+n2 = m22 - n22

= m4m2 - n4n2

(v) (3t + 9s) (3t – 9s)

Solution:

Using (a - b)(a + b) = a22a - b22b, where a = 3t and b = 9s:

(3t + 9s)(3t - 9s) = 3t2 - 9s2 = 9t28t2 - 81s281s3

(vi) (kl – mn) (kl + mn)

Solution:

Using (a - b)(a + b) = a22a - b22b, where a = and b = :

(kl - mn)(kl + mn) = (kl2kl) - (mn2mn)

= k2l22kl - m2n2mn

(vii) (6x + 5) (6x + 6)

Solution:

(6x + 5)(6x + 6) = (6x236x) + (5+6)(6x) + 5×6

= x2 + + .

(viii) (2b – a) (2b + c)

Solution:

(2b - a)(2b + c) = 2b2 + (c-a)(2b) - ac

= b2 + 2bc - -

2. Evaluate the following by using suitable identities:

(i) 3042

Solution:

3042 = 300+42 = 30023042 + 2()() + 424

= + + =

(ii) 5092

Solution:

5092 = 500+92 = 50025092 + 2()() + 9232

= + + =

(iii) 9922

Solution:

9922 = 1000−82 = 100021000 - 2()() + 8242

= - + =

(iv) 7992

Solution:

7992 = 800−12 = 8002801 - 2(800)(1) + 12

= - + =

(v) 304 × 296

Solution:

304 can be written as 300 + and 296 can be written as 300 -

= (300 + 4)(300 - 4) = 3002 - 42

= - =

(vi) 83 × 77

Solution:

83 can be written as 80 + and 77 can be written as 80 -

= (80 + 3)(80 - 3) = 802 - 32

= - =

(vii) 109 × 108

Solution:

109 × 108 = (100 + )(100 + )

= 1002103 + (9)100 + (8)100 + 9×8

= + + + =

(viii) 204 × 206

Solution:

204 × 206 = (200 + )(200 + )

= 2002203 + (4+6)200 + 4×6

= + + =