Cubes and Cube Roots

See The Chapter PDF

Cube Numbers

If a natural number m can be expressed as n³ , where n is also a natural number, then m is called the cube number of n.
Numbers like 1, 8, 27 are cube number of the numbers 1, 2, and 3 respectively.
All perfect cube numbers are obtained by multiplying a number by itself three times.

Cubes Relation with Cube Numbers

In geometry, a cube is a solid figure where all edges are equal and are perpendicular to each other.

For example, take a cube of unit side. If we arrange these cubes to form a bigger cube of side 3 units, we find that there are a total of 27 such unit cubes that make up a cube of 3 units. Similarly, a cube of 4 units will have 64 such unit cubes.

Units Digits in Cube Numbers

Depending on whether a number is odd or even, its cube number is also odd or even respectively.
This is determined by the nature of the cube numbers’ unit digit.

If a number is odd, its cube numbers’ unit digit is also odd.
If a number is even, its cube numbers’ unit digit is also even.

The table below shows the units digit of a number and the units digit of the cube of that number:

Units digit of number	Units digit of its cube
1	1
2	8
3	7
4	4
5	5
6	6
7	3
8	2
9	9

Inside Cube Numbers

Adding Consecutive Odd Numbers

1 = 1 = 1³
3 + 5 = 8 = 2³
7 + 9 + 11 = 27 = 3³
13 + 15 + 17 + 19 = 64 = 4³
21 + 23 + 25 + 27 + 29 = 125 = 5³

We can see from the above pattern, if we need to find the n³ , n consecutive odd numbers will be needed, such that their sum is equal to n³ .
This pattern holds true for all natural numbers.

Also, if we need to find n³ then we should add n consecutive natural numbers starting from n² - n + 1 .

Example: Finding 8³ by expressing it as a sum of consecutive odd numbers.
First term:
= n² - n + 1
= 8² - 8 + 1
= 64 - 8 + 1
= 57
Sum of 8 consecutive odd natural terms from 31
= 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71
= 512
= 8³

Prime Factorisation Method to Find a Cube

In the prime factorisation of any number, if each prime factor appears three times , then the number is a perfect cube .

Consider, the number 216. By prime factorisation,
216 = 2 × 2 × 2 × 3 × 3 × 3
216 = 2³ × 3³
216 = 6³
Hence, 216 is a perfect cube.

Consider, the number 500. By prime factorisation,
500 = 2 × 2 × 5 × 5 × 5
500 = 2² × 5³
In the above prime factorisation 2 appears twice.
Hence, 500 is not a perfect cube.

Cube Roots

Finding the cube root is the inverse operation of finding the cube . We know that 3³ =27. We can also write the same equation as ∛27 =3. The symbol ‘∛‘ denotes ‘ cube root ‘.

Smallest Multiple that is a Perfect Cube

Consider an example: 53240.
Now, we have to check whether the given number 53240 is a perfect cube or not.
So, to find whether the given number is a perfect cube or not, first we have to find the prime factorisation of 53240.
Hence, the prime factorisation of 53240 is 2 × 2 × 2 × 5 × 11 × 11 × 11.
Here, the number 11 is repeated thrice and “2” is also repeated thrice. But we don’t have three 5’s.
Hence, the given number is not a perfect cube.

Thus, to make 53240 a perfect cube, we should multiply by 5 twice on both sides because we need two more factors of 5 to make 5 appear 3 times .
53240 × 25 = 2 × 2 × 2 × 5 × 25(5 × 5) × 11 × 11 × 11 53240 × 25 = 2 × 2 × 2 × 5 × 5 × 5 ×
53240 × 25 = 11 × 11 × 11
So, we have three 2’s, three 5’s and three 11’s.
1331000 = 2³ × 5³ × 11³
1331000 = (2 × 5 × 11)³
1331000 = 110³
Hence, 1331000 is a perfect cube.
Therefore, the smallest natural number by which 53240 must be multiplied to make a perfect cube is 25.

Alternate Method:

As we know, the prime factorisation of 53240 = 2 × 2 × 2 × 5 × 11 × 11 × 11.
So, if we divide 53240 by 5 on both sides, we will get
53240 5 = 2 × 2 × 2 × 5 × 11 × 11 × 11 5
On simplification, we get
10648 = 2 × 2 × 2 × 11 × 11 × 11
Hence, we have three 2’s and three 11’s, and we can say that 10648 is a perfect cube number.
I.e.,
10648 = 2³ × 11³
10648 = (2 × 11)³
10648 = 22³
Therefore, the smallest natural number by which 53240 must be divided to make a perfect cube is 5.

Cube root using prime factorisation

We can find the cube root of a number by prime factorisation method by the following steps:

resolve the number into its prime factors. Consider the number 5832.
5832 = (2 × 2 × 2 ) × ( 3 × 3 × 3 ) × ( 3 × 3 × 3 )
5832 = (2 × 2 × 2 ) × ( 3 × 3 × 3 ) ×
5832 = ( 3 × 3 × 3 )
make groups of three same prime factors.
take one prime factor from each group and multiply them. Their product is the required cube root.

Therefore, cube root of 5832
= ∛5832
= 2 × 3 × 3 = 18

Be More Curious!

Cube Root of a Cube Number using estimation

If a cube number is given we can find out its cube root using the following steps:

Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number. So 117649 has two groups, and first group (649) and the second group (117).
The unit’s digit of the first group (649) will decide the unit digit of the cube root . Since the number 649 ends with 9, the cube roots unit’s digit is 9.
Find the cube of numbers between which the second group lies . The other group is 117. We know that 4³ = 64 and 5³ = 125. 64 < 117 < 125 . Take the smaller number between 4 and 5 as the ten’s digit of the cube root. So, 49 is the cube root of 117649.

Differences of Squares of Triangular Numbers and Converse

Triangular numbers: It is a sequence of the numbers 1, 3, 6, 10, 15 etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles .

Sum of two consecutive triangular numbers gives us a square number . For example,
1 + 3 = 4 = 2² and
3 + 6 = 9 = 3².

The difference between the squares of two consecutive triangular numbers is a cube number . For example,
3² − 1² = 9 − 1 = 8 = 2³ and
6² − 3² = 36 − 9 = 25 = 5³

Also, if the difference between the squares of two numbers is a cube number , then these numbers are consecutive triangular numbers.

Short Answer Questions(SAQs)

What are ‘cube numbers’?

A cube number is a number multiplied by itself 3 times.

What are consecutive triangular numbers?

The sequence of triangular numbers, starting with the 0th triangular number is called the ‘Consecutive triangular numbers’.

What is a Unit digit?

Units digit of a number is the digit in the one’s place of the number.

MCQs

1. What is the cube of 3?

(a) 9

(b) 18

(d) 81

► (c) 27

2. What is ∛64?

(a) 6

(b) 8

(d) 7

► (c) 4

3. Cube of −2 is:

(a) −6

(b) 8

(d) 4

► (c) −8

4. ∛1 equals:

(a) 0

(b) 1

(d) ∞

► (b) 1

5. Cube of 0 is:

(a) 1

(b) 0

(d) −1

► (b) 0

6. Which number is a perfect cube?

(a) 81

(b) 125

(d) 36

► (b) 125

7. Cube of 5 is:

(a) 15

(b) 125

(d) 100

► (b) 125

8. Cube of a negative number is always:

(a) Positive

(b) Negative

(d) Undefined

► (b) Negative

9. ∛1000 =

(a) 100

(b) 50

(d) 20

► (c) 10

10. Which of the following is not a perfect cube?

(a) 27

(b) 64

(d) 125

► (c) 100

11. How many digits can a cube end with?

(a) 0, 1, 4, 5, 6, 9

(b) 1, 8, 7, 0, 3, 6

(d) Only odd digits

► (c) 0 to 9

12. Which of the following cube roots is correct?

(a) ∛343 = 5

(b) ∛729 = 8

(d) ∛216 = 4

► (c) ∛512 = 8

13. Cube root of 216 is:

(a) 4

(b) 5

(d) 7

► (c) 6

14. What is the cube of 1.5?

(a) 2.25

(b) 3.375

(d) 4.125

► (b) 3.375

15. A cube root of a number 'n' is a number that when cubed gives:

(a) n

(b) √n

(d) log(n)

► (a) n

16. Cube of 11 is:

(a) 121

(b) 132

(d) 143

► (c) 1331

17. Cube of even number is always:

(a) Even

(b) Odd

(d) Fraction

► (a) Even

18. What is the cube root of 729?

(a) 8

(b) 9

(d) 3

► (b) 9