🧊 Class 8 · Number system

Cubes and Cube Roots for Class 8: notes, worked examples and practice

A cube number is a number multiplied by itself three times, like the number of unit cubes that fill a bigger cube. A cube root takes you back.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

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Notes

Cubes and Cube Roots: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

Cube numbers

n³ = n × n × n. The first cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

Patterns

The cube of an even number is even and the cube of an odd number is odd. The cube of a negative number is negative: (−4)³ = −64.

Cube roots

The cube root of a number is the number which, multiplied by itself three times, gives it. ∛64 = 4 because 4 × 4 × 4 = 64.

Prime factorisation

Write the number as a product of primes and group equal primes into triplets. Take one from each triplet. 3375 = 3³ × 5³, so ∛3375 = 3 × 5 = 15.

A famous number

1729 is the smallest number that can be written as the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³.

Formulas and rules to remember

n³ = n × n × n
∛(n³) = n
∛(a × b) = ∛a × ∛b
(−n)³ = −n³

Solved examples

Example 1

Find ∛3375 by prime factorisation.

3375 = 3 × 3 × 3 × 5 × 5 × 5.
Take one from each triplet: 3 × 5 = 15.

Answer: 15

Example 2

Is 500 a perfect cube?

500 = 2 × 2 × 5 × 5 × 5.
The prime 2 appears only twice, so it does not form a triplet.
So 500 is not a perfect cube.

Answer: No

Example 3

What is the smallest number by which 108 must be multiplied to get a perfect cube?

108 = 2 × 2 × 3 × 3 × 3.
The prime 2 appears twice, so one more 2 is needed.
108 × 2 = 216 = 6³.

Answer: 2

Example 4

Find ∛(−343).

343 = 7³, so ∛343 = 7.
The cube root of a negative number is negative.

Answer: −7

Avoid these

Common mistakes

Mixing up cube and triple

5³ means 5 × 5 × 5 = 125, not 5 × 3 = 15. The small 3 tells you how many times to multiply.

Splitting a sum under the root

∛(a + b) is not ∛a + ∛b. 1 + 7 = 8 has cube root 2, but ∛1 + ∛7 is not 2.

Dropping the sign for negative numbers

∛(−27) = −3, because (−3)³ = −27.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

13³ =
Answer: 2197
The cube root of 8 is
Answer: 2
3³ =
Answer: 27
7³ =
Answer: 343
10³ =
Answer: 1000
11³ =
Answer: 1331
5² =
Answer: 25
20² =
Answer: 400
15² =
Answer: 225
19² =
Answer: 361

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FAQ

Cubes and Cube Roots Class 8: questions students ask

What is a cube number?

A number multiplied by itself twice, for example 4³ = 4 × 4 × 4 = 64.

How do I find a cube root without a calculator?

Use prime factorisation, make groups of three equal primes and multiply one from each group.

What is the cube of a negative number?

It is negative, for example (−2)³ = −8.

What is special about 1729?

It is the smallest number that is the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³.

Which board is this cubes and cube roots page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

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