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1. Find which of the following numbers is a perfect cube?
(a) 256
(b) 81
(c) 2744
(d) 16
Solution:
(a) 256
Find the Prime Factors of the given number 256
256 prime factors are 4, 4, 4, 4
256 = 4 × 4 × 4 × 4
Form the triplets with the group of the same prime factor numbers.
256 = (4 × 4 × 4) × 4
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.
(b) 81
Find the Prime Factors of the given number 81
81 prime factors are 3, 3, 3, 3
81 = 3 × 3 × 3 × 3
Form the triplets with the group of the same prime factor numbers.
81 = (3 × 3 × 3) × 3
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.
(c) 2744
Find the Prime Factors of the given number 2744
2744 prime factors are 14, 14, 14
2744 = 14 × 14 × 14
Form the triplets with the group of the same prime factor numbers.
2744 = (14 × 14 × 14)
There is only one triplet available
(d) 16
Find the Prime Factors of the given number 16
16 prime factors are 2, 2, 2, 2
16 = 2 × 2 × 2 × 2
Form the triplets with the group of the same prime factor numbers.
16 = (2 × 2 × 2) × 2
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.
Therefore, (c) 2744 is a perfect cube.
2. Which of the following numbers is not a perfect cube?
(a) 1331
(b) 729
(c) 4096
(d) 2744
Solution:
(a) 1331
Find the Prime Factors of the given number 1331
1331 prime factors are 11, 11, 11
1331 = 11 × 11 × 11
Form the triplets with the group of the same prime factor numbers.
1331 = (11 × 11 × 11)
There is only one triplet available.
Therefore, the given number is a perfect cube.
(b) 729
Find the Prime Factors of the given number 729
729 prime factors are 9, 9, 9
729 = 9 × 9 × 9
Form the triplets with the group of the same prime factor numbers.
729 = (9 × 9 × 9)
There is only one triplet available.
Therefore, the given number is a perfect cube.
(c) 4096
Find the Prime Factors of the given number 4096
4096 prime factors are 8, 8, 8, 8
4096 = 8 × 8 × 8 × 8
Form the triplets with the group of the same prime factor numbers.
4096 = (8 × 8 × 8) × 8
There is only one triplet and one single factor available.
Therefore, the given number is not a perfect cube.
(d) 2744
Find the Prime Factors of the given number 2744
2744 prime factors are 14, 14, 14
2744 = 14 × 14 × 14
Form the triplets with the group of the same prime factor numbers.
2744 = (14 × 14 × 14)
There is only one triplet available.
Therefore, the given number is a perfect cube.
(c) 4096 is not a perfect cube.
3. What least number must be multiplied to 8575 so that the product becomes a perfect cube?
(a) 4
(b) 2
(c) 7
(d) 5
Solution:
Find the Prime Factors of the given number 8575
8575 prime factors are 7, 7, 7, 5, 5
8575 = 7 × 7 × 7 × 5 × 5
Form the triplets with the group of the same prime factor numbers.
8575 = (7 × 7 × 7) × (5 × 5)
Multiply 5 to the given number to make it a perfect cube.
4. What is the least number by which 12096 must be divided so that the quotient is a perfect cube?
(a) 5
(b) 7
(c) 3
(d) 4
Solution:
Find the Prime Factors of the given number 12096
12096 prime factors are 12, 12, 12, 7
12096 = 12 × 12 × 12 × 7
Form the triplets with the group of the same prime factor numbers.
12096 = (12 × 12 × 12) × 7
Divide 7 to the given number to make it a perfect cube.
5. ∛4913
(a) 18
(b) 21
(c) 14
(d) 17
Solution:
Write the product of primes of a given number 4913 those form groups in triplets.
Cube Root of 4913 = ∛4913 = ∛(17 ×17 × 17)
Take one number from a group of triplets to find the cube root of 4913.
Therefore, 17 is the cube root of a given number 4913.
(d) 17 is the cube root of 4913
6. Evaluate: ∛[(2197)/216]
(a) 6/13
(b) 12/11
(c) 11/12
(d) 13/6
Solution:
Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(2197/216) = ∛2197/∛216
Then, find the prime factors for each integer separately.
[∛(13 × 13 × 13)]/[ ∛(6 × 6 × 6)]
Take each integer from the group in triplets to get the cube root of a given number.
13/6
13/6 is the cube root of ∛(2197/216).