**Question 1. Find the cube root of each of the following numbers by prime factorization method.**

**(i) 64**

64 = 2 × 2 × 2 × 2 × 2 × 2

By assembling the factors in trio of equal factors, 64 = (2 × 2 × 2) × (2 × 2 × 2)

Therefore, 64 = 2 × 2 = 4

Hence, 4 is cube root of 64.

**(ii) 512**

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

By assembling the factors in trio of equal factors, 512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

Therefore, 512 = 2 × 2 × 2 = 8

Hence, 8 is cube root of 512.

**(iii) 10648**

10648 = 2 × 2 × 2 × 11 × 11 × 11

By assembling the factors in trio of equal factors, 10648 = (2 × 2 × 2) × (11 × 11 × 11)

Therefore, 10648 = 2 × 11 = 22

Hence, 22 is cube root of 10648.

**(iv) 27000**

27000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5

By assembling the factors in trio of equal factors, 27000 = (2 × 2 × 2) × (3 × 3 × 3) × (5 × 5 × 5)

Therefore, 27000 = (2 × 3 × 5) = 30

Hence, 30 is cube root of 27000.

**(v) 15625**

15625 = 5 × 5 × 5 × 5 × 5 × 5

By assembling the factors in trio of equal factors, 15625 = (5 × 5 × 5) × (5 × 5 × 5)

Therefore, 15625 = (5 × 5) = 25

Hence, 25 is cube root of 15625.

**(vi) 13824**

13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

By assembling the factors in trio of equal factors,

13824 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)

Therefore, 13824 = (2 × 2 × 2 × 3) = 24

Hence, 24 is cube root of 13824.

**(vii) 110592**

110592 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

By assembling the factors in trio of equal factors,

110592 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 ×3)

Therefore, 110592 = (2 × 2 × 2 × 2 × 3) = 48

Hence, 48 is cube root of 110592.

**(viii) 46656**

46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

By assembling the factors in trio of equal factors,

46656 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)

Therefore, 46656 = (2 × 2 × 3 × 3) = 36

Hence, 36 is cube root of 46656.

**(ix) 175616**

175616 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7

By assembling the factors in trio of equal factors,

175616 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (7 × 7 × 7)

Therefore, 175616 = (2 × 2 × 2 × 7) = 56

Hence, 56 is cube root of 175616.

**(x) 91125**

91125 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5

By assembling the factors in trio of equal factors, 91125 = (3 × 3 × 3) × (3 × 3 × 3) × (5 × 5 × 5)

Therefore, 91125 = (3 × 3 × 5) = 45

Hence, 45 is cube root of 91125.

**Question. 2. State true or false.**

**(i) Cube of any odd number is even.**

False

**(ii) A perfect cube does not end with two zeros.**

True

**(iii) If square of a number ends with 5, then its cube ends with 25.**

False

**(iv) There is no perfect cube which ends with 8.**

False

**(v) The cube of a two-digit number may be a three-digit number.**

False

**(vi) The cube of a two-digit number may have seven or more digits.**

False

**(vii) The cube of a single digit number may be a single digit number.**

True

**Question. 3. You are told that 1,331 is a perfect cube. Can you guess without factorization what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.**

**(i) 1331**

Since, the unit digit of cube is 1, the unit digit of cube root 1.

Therefore, we get 1 as unit digit of cube root of 1331.

And the ten’s digit of our cube root is taken as the unit place of the smallest number.

As the unit digit of the cube of a number having digit as unit place 1 is 1.

Therefore,

∛1331 = 11

**(ii) 4913**

Since, the unit digit of cube is 3, the unit digit of cube root will be 7.

Therefore, we get 7 as unit digit of the cube root of 4913. We know 13 = 1 and 23 = 8, 1 > 4 > 8.

So, 1 is taken as ten digits of cube root.

Therefore,

∛4913 = 17

**(iii) 12167**

Since, the unit digit of cube is 7, the unit digit of cube root will be 3.

Therefore, 3 is the unit digit of the cube root of 12167 We know 23 = 8 and 33 = 27, 8 > 12 > 27.

So, 2 is taken as ten digits of cube root.

Therefore,

∛12167 = 23

**(iv) 32768**

Since, the unit digit of cube is 8, the unit digit of cube root will be 2

Therefore, 2 is the unit digit of the cube root of 32768. We know 33 = 27 and 43 = 64, 27 > 32 > 64.

So, 3 is taken as ten digits of cube root.

Therefore,

∛32768 = 32