Mathematics
Evaluate the following using suitable identities:
(i) (99)3
(ii) (102)3
(iii) (998)3
Answer
(i) (99)3
We can write it as,
= (100 - 1)3
We know that,
(a - b)3 = (a)3 - (b)3 - 3ab(a - b)
Putting a = 100 , b = 1
= (100)3 - (1)3 - 3 x 100 x 1(100 - 1)
= 1000000 - 1 - 300 x 99
= 1000000 - 1 - 29700
= 970299
Hence, (99)3 = 970299
(ii) (102)3
We can write it as,
= (100 + 2)3
We know that,
(a + b)3 = (a)3 + (b)3 + 3ab(a + b)
Putting a = 100 , b = 2
= (100)3 + (2)3 + 3 x 100 x 2(100 + 2)
= 1000000 + 8 + 600 x 102
= 1000000 + 8 + 61200
= 1061208
Hence, (102)3 = 1061208
(iii) (998)3
We can write it as,
= (1000 - 2)3
We know that,
(a - b)3 = (a)3 - (b)3 - 3ab(a - b)
Putting a = 1000 , b = 2
= (1000)3 - (2)3 - 3 x 1000 x 2(1000 - 2)
= 1000000000 - 8 - 6000 x 998
= 1000000000 - 8 - 5988000
= 994011992
Hence, (998)3 = 994011992