Q1: Find the cube root of each of the following numbers by prime factorisation method.
(i) 64
Sol: 64 = 2 × 2 × 2 × 2 × 2 × 2
By grouping the factors in triplets of equal factors, 64 = (2 × 2) x (2 × 2) × (2 × 2)
Here, 64 can be grouped into triplets of equal factors.
∴ 64 = (2 x 2)3 = 43
Hence, 4 is the cube root of 64.
(ii) 512
Sol: 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
By grouping the factors in triplets of equal factors, 512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)Here, 512 can be grouped into triplets of equal factors.
∴ 512 = (2 × 2 × 2)3 = 83
Hence, 8 is the cube root of 512.
(iii) 10648
Sol: 10648 = 2 × 2 × 2 × 11 × 11 × 11
By grouping the factors in triplets of equal factors, 10648 = (2 × 2 × 2) × (11 × 11 × 11)Here, 10648 can be grouped into triplets of equal factors.
∴ 10648 = (2 × 2 x 2) x (11 x 11 x 11) = 23 x 113 = (22)3
Hence, 22 is the cube root of 10648.
(iv) 27000
Sol: 27000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5
By grouping the factors in triplets of equal factors, 27000 = (2 × 2 × 2) × (3 × 3 × 3) × (5 × 5 × 5)Here, 27000 can be grouped into triplets of equal factors.
∴ 27000 = (2 × 3 × 5)3 = 303
Hence, 30 is the cube root of 27000.
(v) 15625
Sol: 15625 = 5 × 5 × 5 × 5 × 5 × 5
By grouping the factors in triplets of equal factors, 15625 = (5 × 5 × 5) × (5 × 5 × 5)Here, 15625 can be grouped into triplets of equal factors.
∴ 15625 = (5 × 5)3 = 253
Hence, 25 is the cube root of 15625.
(vi) 13824
Sol: 13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
By grouping the factors in triplets of equal factors,
13824 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)Here, 13824 can be grouped into triplets of equal factors.
∴ 13824 = (2 × 2 × 2 × 3)3 = 243
Hence, 24 is the cube root of 13824.
(vii) 110592
Sol: 110592 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
By grouping the factors in triplets of equal factors,
110592 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)Here, 110592 can be grouped into triplets of equal factors.
∴ 110592 = (2 × 2 × 2 × 2 × 3)3 = 483
Hence, 48 is the cube root of 110592.
(viii) 46656
Sol: 46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
By grouping the factors in triplets of equal factors,
46656 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)Here, 46656 can be grouped into triplets of equal factors.
∴ 46656 = (2 × 2 × 3 × 3)3 = 363
Hence, 36 is the cube root of 46656.
(ix) 175616
Sol: 175616 = 2×2×2×2×2×2×2×2×2×7×7×7
By grouping the factors in triplets of equal factors,
175616 = (2×2×2)×(2×2×2)×(2×2×2)×(7×7×7)Here, 175616 can be grouped into triplets of equal factors.
∴ 175616 = (2×2×2×7)3 = 563
Hence, 56 is the cube root of 175616.
(x) 91125
Sol: 91125 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5
By grouping the factors in triplets of equal factors, 91125 = (3 × 3 × 3) × (3 × 3 × 3) × (5 × 5 × 5)Here, 91125 can be grouped into triplets of equal factors.
∴ 91125 = (3 × 3 × 5)3 = 453
Hence, 45 is the cube root of 91125.
Q2: State True or False
(i) Cube of any odd number is even.
Ans: False
(ii) A perfect cube does not end with two zeros.
Ans: True
(iii) If square of a number ends with 5, then its cube ends with 25.
Ans: False
(iv) There is no perfect cube which ends with 8.
Ans: False
(v) The cube of a two digit number may be a three digit number.
Ans: False
(vi) The cube of a two digit number may have seven or more digits.
Ans: False
(vii) The cube of a single digit number may be a single digit number.
Ans: True
Question: You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
Solution:
(i) Separating the given number (1331) into two groups:
1331 → 1 and 331
∵ 331 end in 1.
∴ Unit’s digit of the cube root = 1
∵ 13 = 1 and ∛1 = 1
∴ Ten’s digit of the cube root = 1
∴ ∛1331 = 11
(ii) Separating the given number (4913) in two groups:
4913 → 4 and 913
Unit’s digits:
∵ Unit’s digit in 913 is 3.
∴ Unit’s digit of the cube root = 7
[73 = 343; which ends in 3]
Ten’s digit:
∵ 13 = 1, 23 = 8
and 1 < 4 < 8
i.e. 13 < 4 < 23
∴ The ten’s digit of the cube root is 1.
∴ ∛4913 = 17
(iii) Separating 12167 in two groups:
12167 → 12 and 167
Unit’s digit:
∵ 167 is ending in 7 and cube of a number ending in 3 ends in 7.
∴ The unit’s digit of the cube root = 3
Ten’s digit:
∵ 23 = 8 and 33 = 27
Also, 8 < 12 < 27
or 23 < 12 < 32
∴ The tens digit of the cube root can be 2.
Thus, ∛12167 = 23.
(iv) Separating 32768 in two groups:
32768 → 32 and 786
Unit’s digit:
768 will guess the unit’s digit in the cube root.
∵ 768 ends in 8.
∴ Unit’s digit in the cube root = 2
Ten’s digit:
∵ 33 = 27 and 43 – 64
Also, 27 < 32 < 64
or 33 < 32 < 43
∴ The ten’s digit of the cube root = 3.
Thus, ∛32768 = 32
79 videos|408 docs|31 tests
|
1. What is a cube and what are its properties? |
2. How to calculate the volume of a cube? |
3. What is a cube root and how to find it? |
4. How to find the surface area of a cube? |
5. How are cubes used in real-life applications? |
|
Explore Courses for Class 8 exam
|