Evaluate each of the following :
(i) 111³-89³
(ii) 46³+34³
(iii) 104³+96³
(iv) 93³-107³
Answers
(i) Given : 111³ - 89³
111³ - 89³ = (100 + 11)³ - (100 - 11)³
By using the identity : (a + b)³ - (a + b)³ = 2(b³ + 3a²b)
Here a = 100 & b = 11
111³ - 89³ = 2(11³ + 3 × 100² × 11)
111³ - 89³ = 2(1331 + 3 × 10000 × 11)
111³ - 89³ = 2(1331 + 330000)
111³ - 89³ = 2 × 331331
111³ - 89³ = 662662
Hence, 111³ - 89³ is 662662.
(ii) Given : 46³ + 34³
46³ + 34³ = (40 + 6)³ + (40 - 6)³
By using the identity : (a + b)³ + (a + b)³ = 2(a³ + 3ab²)
Here a = 40 & b = 6
46³ + 34³ = 2(40³ + 3 × 40 × 6²)
46³ + 34³ = 2(64000 + 3 × 40 × 36)
46³ + 34³ = 2(64000 + 4320)
46³ + 34³ = 2 × 68320
46³ + 34³ = 136640
Hence, 104³ + 96³ is 2009600.
(iii) Given : 104³ + 96³
104³ + 96³ = (100 + 4)³ + (100 - 4)³
By using the identity : (a + b)³ + (a + b)³ = 2(a³ + 3ab²)
Here a = 100 & b = 4
104³ + 96³ = 2(100³ + 3 × 100 × 4²)
104³ + 96³ = 2(1000000 + 3 × 100 × 16)
104³ + 96³ = 2(1000000 + 4800)
104³ + 96³ = 2 × 1004800
104³ + 96³ = 2009600
Hence, 104³ + 96³ is 2009600.
(iv) Given : 93³ - 107³
93³ - 107³ = (100 - 7)³ - (100 + 7)³
By using the identity : (a + b)³ - (a + b)³ = 2(b³ + 3a²b)
Here a = 100 & b = 7
93³ - 107³ = - 2(7³ + 3 × 100² × 7)
93³ - 107³ = - 2(343 + 3 × 10000 × 7)
93³ - 107³ = - 2(343 + 210000)
93³ - 107³ = - 2 × 210343
93³ - 107³ = - 420686
Hence, 93³ - 107³ is - 420686.
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Answer:
Step-by-step explanation:
(i) Given : 111³ - 89³
111³ - 89³ = (100 + 11)³ - (100 - 11)³
By using the identity : (a + b)³ - (a + b)³ = 2(b³ + 3a²b)
Here a = 100 & b = 11
111³ - 89³ = 2(11³ + 3 × 100² × 11)
111³ - 89³ = 2(1331 + 3 × 10000 × 11)
111³ - 89³ = 2(1331 + 330000)
111³ - 89³ = 2 × 331331
111³ - 89³ = 662662
(ii) Given : 46³ + 34³
46³ + 34³ = (40 + 6)³ + (40 - 6)³
By using the identity : (a + b)³ + (a + b)³ = 2(a³ + 3ab²)
Here a = 40 & b = 6
46³ + 34³ = 2(40³ + 3 × 40 × 6²)
46³ + 34³ = 2(64000 + 3 × 40 × 36)
46³ + 34³ = 2(64000 + 4320)
46³ + 34³ = 2 × 68320
46³ + 34³ = 136640
(iii) Given : 104³ + 96³
104³ + 96³ = (100 + 4)³ + (100 - 4)³
By using the identity : (a + b)³ + (a + b)³ = 2(a³ + 3ab²)
Here a = 100 & b = 4
104³ + 96³ = 2(100³ + 3 × 100 × 4²)
104³ + 96³ = 2(1000000 + 3 × 100 × 16)
104³ + 96³ = 2(1000000 + 4800)
104³ + 96³ = 2 × 1004800
104³ + 96³ = 2009600