Cubing Numbers Using Suitable Identities

To evaluate the given expressions, we can use suitable identities for cubing numbers. These identities are:

1. (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
2. (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Using these identities, we can simplify the expressions as follows:

(i) (99)^3
We can write 99 as (100 - 1). Using the second identity, we get:
(100 - 1)^3 = 100^3 - 3(100^2)(1) + 3(100)(1^2) - 1^3
= 1000000 - 30000 + 300 - 1
= 970299

(ii) (102)^3
We can write 102 as (100 + 2). Using the first identity, we get:
(100 + 2)^3 = 100^3 + 3(100^2)(2) + 3(100)(2^2) + 2^3
= 1000000 + 60000 + 1200 + 8
= 1061208

(iii) (998)^3
We can write 998 as (1000 - 2). Using the second identity, we get:
(1000 - 2)^3 = 1000^3 - 3(1000^2)(2) + 3(1000)(2^2) - 2^3
= 1000000000 - 6000000 + 12000 - 8
= 999988992

Therefore, the values of the given expressions are:
(i) (99)^3 = 970299
(ii) (102)^3 = 1061208
(iii) (998)^3 = 999988992