Evaluate the following using suitable identities:

(i)  (99)³ = (100 – 1)³

Using identity,

(a – b)³ = a³ – b³ – 3ab (a – b)

(100 – 1)³ = (100)³ – 1³ – (3 * 100 * 1) (100 – 1)

 = 1000000 – 1 – 300 (100 – 1)

 = 1000000 – 1 – 30000 + 300

 = 970299

(ii) (102)³ = (100 + 2)³

Using identity,

(a – b)³ = a³ – b³ – 3ab (a – b)

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

 = 1000000 + 8 + 600 (100 + 2)

 = 1000000 + 8 + 60000 + 1200

 = 1061208

(iii)  (998)³ = (1000 – 2)³

Using identity,

(a – b)³ = a³ – b³ – 3ab (a – b)

(1000 – 2)³ = (1000)³ – 2³ – (3 * 1000 * 2) (1000 – 2)

= 100000000 – 8 – 6000 (1000 – 2)

= 100000000 – 8 – 6000000 + 12000

= 994011992