Points to Remember - Cubes and Cube Roots | Mathematics (Maths) Class 8 PDF Download

Facts That Matter

• If we multiply a number by itself three times, the product so obtained is called the perfect cube of that number.
• There are only 10 perfect cubes from 1 to 1000.
• Cubes of even numbers are even and those of odd numbers are odd.
• The cube of a negative number is always negative

• If the digit in the one’s place of a number is 0, 1, 4, 5, 6 or 9, then its cube will end in the same digit.
• If the digit in the one’s place of a number is 2, then the ending digit of its cube will be 8 and vice-versa.
• If the digit in the one’s place of a number is 3, then the ending digit of its cube will be 7 and vice-versa.
• If the prime factors of a number cannot be made into groups of 3, it is not a perfect cube.
• The symbol ∛ denotes the cube root of a number, such as ∛64 = 4.

Steps to Calculate Cube Root

(i) The square of a number is obtained by multiplying the number by itself, i.e.
2 * 2 = 2²
4 * 4 = 4²
5 * 5 = 5²

(ii) Finding square root is the inverse operation of finding the square of a number, i.e.

(iii) A natural number multiplied by itself three times gives a cube of that number, e.g.
1 * 1 * 1 = 1
2 * 2 * 2 = 8
3 * 3 * 3 = 27
4 * 4 * 4 = 64
The numbers 1, 8, 27, 64, … are called cube numbers or perfect cubes.

Properties of Perfect Cubes

(a) Property 1

(b) Property 2

If the digit in the one’s place of a number is 2, the digit in the one’s place of its cube is 8, and vice-versa.

(c) Property 3

If the digit in the one’s place of a number is 3, the digit in the one’s place of its cube is 7 and vice-versa.

Examples of Properties 1, 2 & 3

(d) Property 4 

Cubes of even natural numbers are even.

Examples of Property 4

(e) Property 5

Cubes of odd natural numbers are odd.
Examples of Property 5

(f) Property 6

Cubes of negative integers are negative.

Examples of Property 6

Some Interesting Patterns in Cubes

1. Adding consecutive odd numbers

Note that we start with [n * (n – 1) + 1] odd number.

2. Difference of two consecutive cubes:

2³ – 1³ = 1 + 2 * 1 * 3

3³ – 2³ = 1 + 3 * 2 * 3

4³ – 3³ = 1 + 4 * 3 * 3

5³ – 4³ = 1 + 5 * 4 * 3

3. Cubes and their prime factor

Each prime factor of the number appears three times in its cube.

Solved Examples

Example 1: Is 500 a perfect cube?
Solution: 

500 = 5 * 5 * 5 * 2 * 2
∵ In the above prime factorisation 2 * 2 remain after grouping the prime factors in triples.
∴ 500 is not a perfect cube.

Example 2: Is 1372 a perfect cube? If not, find the smallest natural number by which 1372 must be multiplied so that the product is a perfect cube.
Solution: 

We have 1372 = 2 * 2 * 7 * 7 * 7
Since, the prime factor 2 does not appear in a group of triples.
∴ 1372 is not a perfect cube.
Obviously, to make it a perfect cube we need one more 2 as its factor.
i.e. [1372] * 2 = [2 * 2 * 7 * 7 * 7] * 2
or
2744 = 2 * 2 * 2 * 7 * 7 * 7
which is a perfect cube.
Thus, the required smallest number = 2.

Example 3: Is 31944 a perfect cube? If not then by which smallest natural number should 31944 be divided so that the quotient is a perfect cube?
Solution: 

We have 31944 = 2 * 2 * 2 * 3 * 11 * 11 * 11
Since, the prime factors of 31944 do not appear in triples as 3 is left over.
∴ 31944 is not a perfect cube. Obviously, 31944 / 3 will be a perfect cube
i.e. [31944] ÷3 = [2 * 2 * 2 * 3 * 11 * 11 * 11] /3
or
10648 = 2 * 2 * 2 * 11 * 11 * 11
∴ 10648 is a perfect cube.
Thus, the required least number = 3.