Q.7. Evaluate the following using suitable identities:

(i) (99)3 (ii) (102)3 (iii) (998)3

Solution:

(i) We can write, (99)3 = (100 - 1)3

Using identity, (a - b)3 = a3 - b3 - 3ab(a - b),

(100 - 1)= (100)3 - 13 - (3×100×1)(100 - 1)

= 1000000 - 1 - 300(100 - 1)

= 1000000 - 1 - 30000 + 300

= 970299

(ii) We can write, (102)3 = (100 + 2)3

Using identity, (a + b)3 = a3 + b3 + 3ab(a + b),

(100 + 2)= (100)3 + 23 + (3×100×2)(100 + 2)

= 1000000 + 8 + 600(100 + 2)

= 1000000 + 8 + 60000 + 1200

= 1061208

(iii)We can write, (998)3 = (1000 - 2)

Using identity, (a - b)3 = a3 - b3 - 3ab(a - b)

(1000-2)= (1000)3 - 23 - (3×1000×2)(1000 - 2)

= 100000000 - 8 - 6000(1000 - 2)

= 100000000 - 8- 600000 + 12000

= 994011992