Simplification
- The cube of 997 is :
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(997)3 = (1000 – 3)3
= (1000)3 – (3)3 – 3 × 1000 × 3 (1000 – 3)
= 1000000000 – 27 – 9000 × 997
= 1000000000 – 27 – 8973000
= 991026973Correct Option: A
(997)3 = (1000 – 3)3
= (1000)3 – (3)3 – 3 × 1000 × 3 (1000 – 3)
= 1000000000 – 27 – 9000 × 997
= 1000000000 – 27 – 8973000
= 991026973
- The least number which when divides 37044, gives the result a perfect cube, is :
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∴ 37044 = 3 × 3 × 3 × 7 × 7 × 7 × 2 × 2
= 33 × 73 × 22
∴ Required number = 2 × 2 = 4Correct Option: B
∴ 37044 = 3 × 3 × 3 × 7 × 7 × 7 × 2 × 2
= 33 × 73 × 22
∴ Required number = 2 × 2 = 4
- The cube of 105 is
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(105)3 = (100 + 5)3
= (100)3 + (5)3 + 3 × 100 × 5 (100 + 5)
[∵ (a + b)3 = a3 + b3 + 3ab (a + b)]
= 1000000 + 125 + 1500 × 105
= 1000000 + 125 + 157500
= 1157625Correct Option: D
(105)3 = (100 + 5)3
= (100)3 + (5)3 + 3 × 100 × 5 (100 + 5)
[∵ (a + b)3 = a3 + b3 + 3ab (a + b)]
= 1000000 + 125 + 1500 × 105
= 1000000 + 125 + 157500
= 1157625
- Find the cube root of (–13824).
or
Find the value of 3√−13824.
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∴ 13824 = 23 × 23 × 23 × 33
∴ 3√–13824
= 3√(−1)3 23 × 23 × 23 × 33
= (–1) 2 × 2 × 2 × 3 = –24Correct Option: D
∴ 13824 = 23 × 23 × 23 × 33
∴ 3√–13824
= 3√(−1)3 23 × 23 × 23 × 33
= (–1) 2 × 2 × 2 × 3 = –24
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When simplified, the product 
2 − 1 
2 − 3 2 − 5 .... 2 − 997 equals 3 5 7 999
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Expression
= 2 − 1 2 − 3 2 − 5 .... 2 − 997 3 5 7 999 = 6 − 1 10 − 3 14 − 5 .... 1998 − 997 3 5 7 999 = 5 × 7 × 9 ×....× 1001 3 5 7 999 = 1001 3 Correct Option: D
Expression
= 2 − 1 2 − 3 2 − 5 .... 2 − 997 3 5 7 999 = 6 − 1 10 − 3 14 − 5 .... 1998 − 997 3 5 7 999 = 5 × 7 × 9 ×....× 1001 3 5 7 999 = 1001 3
