NCERT Solutions for Class 8 Maths - Cubes and Cube Roots - 2 (Exercise 6.2)

Exercise 6.2

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

Deleted Questions from NCERT

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