Cube Root: Questions and Solutions

Cube root questions with answers present methods and worked examples for finding the cube root of numbers through simple and effective steps. This guide reviews the standard idea that the cube root of a number is a value that, when multiplied by itself three times, gives the original number, and demonstrates its use through solved problems involving perfect cubes, prime factorization, and estimation. From straightforward calculations to application-based questions, each solution focuses on clear steps, number reasoning, and helpful shortcuts. Worked examples with brief explanations help strengthen understanding and exam preparation.

Table of Contents

• Methods to Find the Cube Root of a Number
• Solved Examples on Cube Root
• Practice Questions on Cube Root
• Frequently Asked Questions of Cube Root Questions

Methods to Find the Cube Root of a Number

1. Prime Factorization Method

• Write the prime factorization of the given number.

• Group the identical prime factors into sets of three.

• Pick one factor from each group and multiply them together, that product is the cube root.

2. Estimation Method

• Starting from the right, split the digits into groups of three.

• Look at the last digit of the rightmost group to fix the unit digit of the answer, using the cube-ending pattern (for instance, a cube ending in 7 always comes from a number ending in 3).

• Look at the remaining left-hand group and find which two consecutive perfect cubes it falls between, the smaller one's root gives the tens digit.

Solved Examples on Cube Root 

Example 1: Find the cube root of 15625 using prime factorization.

Solution:

15625 = 5 × 5 × 5 × 5 × 5 × 5

Grouping in threes: 15625 = (5 × 5 × 5) × (5 × 5 × 5) = 5³ × 5³

∛15625 = 5 × 5 = 25

Example 2: Find the cube root of 175616 by estimation (given that it is a perfect cube).

Solution:

Group from the right: (175)(616)

The last digit of 616 is 6, and a cube ending in 6 always comes from a number ending in 6 → unit digit = 6

The left group 175 lies between 5³ = 125 and 6³ = 216, so the tens digit is the smaller one, 5

∛175616 = 56  (check: 56³ = 56 × 56 × 56 = 175616 )

Example 3: Find the cube root of 0.004913.

Solution:

0.004913 = 4913/1000000 = 17³/100³

∛0.004913 = 17/100 = 0.17

Example 4: Find the smallest number by which 72 must be multiplied to make it a perfect cube, and find the cube root of the new number.

Solution:

72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

The factor 3 appears only twice; one more 3 is needed to complete a group of three.

Smallest number to multiply = 3

New number = 72 × 3 = 216 = 6³, so its cube root = 6

Example 5: Find the value of 12³ + 7³ + (−19)³.

Solution:

12 + 7 + (−19) = 0. Whenever a + b + c = 0, the identity a³ + b³ + c³ = 3abc applies.

3abc = 3 × 12 × 7 × (−19) = 3 × (−1596) = −4788

Check by direct calculation: 1728 + 343 − 6859 = −4788

Example 6: A cube-shaped water storage tank has a volume of 3375 cubic metres. Find the length of each edge of the tank.

Solution:

Volume of a cube = (side)³, so side = ∛3375

3375 = 15 × 15 × 15 = 15³

Length of each edge = 15 m

Example 7: Is 1728 a perfect cube? If so, find its cube root.

Solution:

1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 2³ × 3³

Since every prime factor's power is a multiple of 3, 1728 is a perfect cube.

∛1728 = 2 × 2 × 3 = 12

Example 8: Assertion (A): The smallest number by which 88 must be multiplied to make it a perfect cube is 121.

Reason (R): 88 = 2³ × 11, and multiplying by 11² completes the group of three for the factor 11.

Solution:

Both A and R are true, and R correctly explains A.

88 × 121 = 10648 = 22³, confirming the new number is a perfect cube. 

Example 9: Simplify: ∛(64 × 343) ÷ ∛(8 × 27)

Solution: 

∛64 = 4, ∛343 = 7, so ∛(64×343) = 4 × 7 = 28

∛8 = 2, ∛27 = 3, so ∛(8×27) = 2 × 3 = 6

Therefore,  ∛(64 × 343) ÷ ∛(8 × 27) = 28 ÷ 6 = 14/3

Example 10: A toy company packs small cubical blocks, each of side 2 cm, tightly into a bigger cube-shaped box of volume 2744 cm³. Find 

(a) the edge of the box, and (b) the total number of small cubes that fit inside.

Solution: 

(a) Edge of box = ∛2744 = ∛(14×14×14) = 14 cm

(b) Number of small cubes along one edge = 14 ÷ 2 = 7

Total small cubes = 7³ = 343 cubes

Example 11: Find the smallest number by which 250 must be divided so that the quotient is a perfect cube. Also find the cube root of the quotient.

Solution: 

250 = 2 × 5 × 5 × 5 = 2¹ × 5³

The lone factor 2 must be removed, so divide by 2.

Quotient = 125 = 5³, so its cube root = 5

Practice Questions on Cube Root

Q1. What is the cube root of 4096?

Q2. Which of these is NOT a perfect cube?

1. 125

2. 216

3. 300

4. 343

Q3. What is the cube root of −1331?

Q4. By which smallest number should 81 be divided to get a perfect cube?

Q5. What is the cube root of 0.000216?

Q6. Assertion (A): The cube root of 9261 is 21.

Reason (R): 21³ = 9261.

Q7. The volume of a cubical storage box is 15625 cm³. Find the length of its edge.

Q8. Find the cube root of 42875 using prime factorization.