NCERT Textbook: Cubes & Cube Roots | Mathematics (Maths) Class 8 PDF Download

 Page 1


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6.1  Introduction
This is a story about one of  India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
interesting. He said it is the smallest  number that can be expressed
as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12
3
 + 1
3
1729 = 1000 + 729 = 10
3
 + 9
3
1729 has since been known as the Hardy – Ramanujan Number,
even though this feature of 1729 was known more than 300 years
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
6.2  Cubes
You know that the word ‘cube’ is used in geometry . A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
taking it three times.
?????? ???? ????? ?????
? ? ? ? ? ? ?
?
Hardy – Ramanujan
Number
1729 is the smallest Hardy–
Ramanujan Number. There
are an infinitely many such
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
2, 24), Check it with the
numbers given in the brackets.
Figures which have
3-dimensions are known as
solid figures.
Reprint 2024-25
Page 2


? ? ? ? ? ? ????? ? ? ???? ? ? ? ? ? ? ?
6.1  Introduction
This is a story about one of  India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
interesting. He said it is the smallest  number that can be expressed
as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12
3
 + 1
3
1729 = 1000 + 729 = 10
3
 + 9
3
1729 has since been known as the Hardy – Ramanujan Number,
even though this feature of 1729 was known more than 300 years
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
6.2  Cubes
You know that the word ‘cube’ is used in geometry . A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
taking it three times.
?????? ???? ????? ?????
? ? ? ? ? ? ?
?
Hardy – Ramanujan
Number
1729 is the smallest Hardy–
Ramanujan Number. There
are an infinitely many such
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
2, 24), Check it with the
numbers given in the brackets.
Figures which have
3-dimensions are known as
solid figures.
Reprint 2024-25
??? ? ? ? ????? ????
The numbers 729, 1000, 1728
are also perfect cubes.
We note that 1 = 1 × 1 × 1 = 1
3
;  8 = 2 × 2 × 2 = 2
3
; 27 = 3 × 3 × 3 = 3
3
.
Since 5
3
 = 5 × 5 × 5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9 = 3 × 3 and there is no natural number which multiplied
by taking three times gives 9. We can see also that 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27. This
shows that 9 is not a perfect cube.
The following are the cubes of numbers from 1 to 10.
Table 1
Number Cube
1 1
3
 = 1
2 2
3
 = 8
3 3
3
 = 27
4 4
3 
= 64
 5 5
3
 = ____
6 6
3
 = ____
7 7
3
 = ____
8 8
3
 = ____
9 9
3
 = ____
10 10
3
 = ____
There are only ten perfect cubes from 1 to 1000. (Check this). How many perfect
cubes are there from 1 to 100?
Observe the cubes of even numbers. Are they all even? What can you say about the
cubes of odd numbers?
Following are the cubes of the numbers from 1 1 to 20.
Table 2
Number Cube
11 1331
12 1728
13 2197
14 2744
15 3375
16 4096
17 4913
18 5832
19 6859
20 8000
We are odd so are
our cubes
We are even, so
are our cubes
Complete it.
Reprint 2024-25
Page 3


? ? ? ? ? ? ????? ? ? ???? ? ? ? ? ? ? ?
6.1  Introduction
This is a story about one of  India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
interesting. He said it is the smallest  number that can be expressed
as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12
3
 + 1
3
1729 = 1000 + 729 = 10
3
 + 9
3
1729 has since been known as the Hardy – Ramanujan Number,
even though this feature of 1729 was known more than 300 years
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
6.2  Cubes
You know that the word ‘cube’ is used in geometry . A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
taking it three times.
?????? ???? ????? ?????
? ? ? ? ? ? ?
?
Hardy – Ramanujan
Number
1729 is the smallest Hardy–
Ramanujan Number. There
are an infinitely many such
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
2, 24), Check it with the
numbers given in the brackets.
Figures which have
3-dimensions are known as
solid figures.
Reprint 2024-25
??? ? ? ? ????? ????
The numbers 729, 1000, 1728
are also perfect cubes.
We note that 1 = 1 × 1 × 1 = 1
3
;  8 = 2 × 2 × 2 = 2
3
; 27 = 3 × 3 × 3 = 3
3
.
Since 5
3
 = 5 × 5 × 5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9 = 3 × 3 and there is no natural number which multiplied
by taking three times gives 9. We can see also that 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27. This
shows that 9 is not a perfect cube.
The following are the cubes of numbers from 1 to 10.
Table 1
Number Cube
1 1
3
 = 1
2 2
3
 = 8
3 3
3
 = 27
4 4
3 
= 64
 5 5
3
 = ____
6 6
3
 = ____
7 7
3
 = ____
8 8
3
 = ____
9 9
3
 = ____
10 10
3
 = ____
There are only ten perfect cubes from 1 to 1000. (Check this). How many perfect
cubes are there from 1 to 100?
Observe the cubes of even numbers. Are they all even? What can you say about the
cubes of odd numbers?
Following are the cubes of the numbers from 1 1 to 20.
Table 2
Number Cube
11 1331
12 1728
13 2197
14 2744
15 3375
16 4096
17 4913
18 5832
19 6859
20 8000
We are odd so are
our cubes
We are even, so
are our cubes
Complete it.
Reprint 2024-25
? ? ? ? ? ? ?????? ? ? ??? ? ? ? ? ? ? ? ?
TRY THESE
each prime factor
appears three times
in its cubes
TRY THESE
Consider a few numbers having 1 as the one’ s digit (or unit’ s). Find the cube of each
of them. What can you say about the one’s digit of the cube of a number having 1 as the
one’s digit?
Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc.
Find the one’s digit of the cube of each of the following numbers.
(i) 3331 (ii) 8888 (iii) 149 (iv) 1005
(v) 1024 (vi) 77 (vii) 5022 (viii) 53
6.2.1  Some interesting patterns
1. Adding consecutive odd numbers
Observe the following pattern of sums of odd numbers.
1 = 1= 1
3
3+ 5= 8 = 2
3
7+ 9+ 11 = 27 = 3
3
13 + 15 + 17 + 19 = 64 = 4
3
21 + 23 + 25 + 27 + 29 = 125 = 5
3
Is it not interesting? How many consecutive odd numbers will be needed to obtain
the sum as 10
3
?
Express the following numbers as the sum of odd numbers using the above pattern?
(a) 6
3
(b) 8
3
(c) 7
3
Consider the following pattern.
2
3
 – 1
3
 = 1 + 2 × 1 × 3
3
3
 – 2
3
 = 1 + 3 × 2 × 3
4
3
 – 3
3
 = 1 + 4 × 3 × 3
Using the above pattern, find the value of the following.
(i) 7
3
 – 6
3
(ii) 12
3
 – 11
3
(iii) 20
3
 – 19
3
(iv) 51
3
 – 50
3
2. Cubes and their prime factors
Consider the following prime factorisation of the numbers and their cubes.
Prime factorisation Prime factorisation
of a number of its cube
4 = 2 × 2 4
3
 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
3
 × 2
3
6 = 2 × 3 6
3
 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2
3
 × 3
3
15 = 3 × 5 15
3
 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3
3
 × 5
3
12 = 2 × 2 × 3 12
3
 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
=2
3
 × 2
3
 × 3
3
Reprint 2024-25
Page 4


? ? ? ? ? ? ????? ? ? ???? ? ? ? ? ? ? ?
6.1  Introduction
This is a story about one of  India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
interesting. He said it is the smallest  number that can be expressed
as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12
3
 + 1
3
1729 = 1000 + 729 = 10
3
 + 9
3
1729 has since been known as the Hardy – Ramanujan Number,
even though this feature of 1729 was known more than 300 years
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
6.2  Cubes
You know that the word ‘cube’ is used in geometry . A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
taking it three times.
?????? ???? ????? ?????
? ? ? ? ? ? ?
?
Hardy – Ramanujan
Number
1729 is the smallest Hardy–
Ramanujan Number. There
are an infinitely many such
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
2, 24), Check it with the
numbers given in the brackets.
Figures which have
3-dimensions are known as
solid figures.
Reprint 2024-25
??? ? ? ? ????? ????
The numbers 729, 1000, 1728
are also perfect cubes.
We note that 1 = 1 × 1 × 1 = 1
3
;  8 = 2 × 2 × 2 = 2
3
; 27 = 3 × 3 × 3 = 3
3
.
Since 5
3
 = 5 × 5 × 5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9 = 3 × 3 and there is no natural number which multiplied
by taking three times gives 9. We can see also that 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27. This
shows that 9 is not a perfect cube.
The following are the cubes of numbers from 1 to 10.
Table 1
Number Cube
1 1
3
 = 1
2 2
3
 = 8
3 3
3
 = 27
4 4
3 
= 64
 5 5
3
 = ____
6 6
3
 = ____
7 7
3
 = ____
8 8
3
 = ____
9 9
3
 = ____
10 10
3
 = ____
There are only ten perfect cubes from 1 to 1000. (Check this). How many perfect
cubes are there from 1 to 100?
Observe the cubes of even numbers. Are they all even? What can you say about the
cubes of odd numbers?
Following are the cubes of the numbers from 1 1 to 20.
Table 2
Number Cube
11 1331
12 1728
13 2197
14 2744
15 3375
16 4096
17 4913
18 5832
19 6859
20 8000
We are odd so are
our cubes
We are even, so
are our cubes
Complete it.
Reprint 2024-25
? ? ? ? ? ? ?????? ? ? ??? ? ? ? ? ? ? ? ?
TRY THESE
each prime factor
appears three times
in its cubes
TRY THESE
Consider a few numbers having 1 as the one’ s digit (or unit’ s). Find the cube of each
of them. What can you say about the one’s digit of the cube of a number having 1 as the
one’s digit?
Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc.
Find the one’s digit of the cube of each of the following numbers.
(i) 3331 (ii) 8888 (iii) 149 (iv) 1005
(v) 1024 (vi) 77 (vii) 5022 (viii) 53
6.2.1  Some interesting patterns
1. Adding consecutive odd numbers
Observe the following pattern of sums of odd numbers.
1 = 1= 1
3
3+ 5= 8 = 2
3
7+ 9+ 11 = 27 = 3
3
13 + 15 + 17 + 19 = 64 = 4
3
21 + 23 + 25 + 27 + 29 = 125 = 5
3
Is it not interesting? How many consecutive odd numbers will be needed to obtain
the sum as 10
3
?
Express the following numbers as the sum of odd numbers using the above pattern?
(a) 6
3
(b) 8
3
(c) 7
3
Consider the following pattern.
2
3
 – 1
3
 = 1 + 2 × 1 × 3
3
3
 – 2
3
 = 1 + 3 × 2 × 3
4
3
 – 3
3
 = 1 + 4 × 3 × 3
Using the above pattern, find the value of the following.
(i) 7
3
 – 6
3
(ii) 12
3
 – 11
3
(iii) 20
3
 – 19
3
(iv) 51
3
 – 50
3
2. Cubes and their prime factors
Consider the following prime factorisation of the numbers and their cubes.
Prime factorisation Prime factorisation
of a number of its cube
4 = 2 × 2 4
3
 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
3
 × 2
3
6 = 2 × 3 6
3
 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2
3
 × 3
3
15 = 3 × 5 15
3
 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3
3
 × 5
3
12 = 2 × 2 × 3 12
3
 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
=2
3
 × 2
3
 × 3
3
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
Observe that each prime factor of a number appears
three times in the prime factorisation of its cube.
In the prime factorisation of any number , if each factor
appears three times, then, is the number a perfect cube?
Think about it. Is 216 a perfect cube?
By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3
Each factor appears 3 times. 216 = 2
3
 × 3
3
 = (2 × 3)
3
= 6
3
   which is a perfect cube!
Is 729 a perfect cube? 729 = 3 × 3 × 3 × 3 × 3 × 3
Yes, 729 is a perfect cube.
Now let us check for 500.
Prime factorisation of 500 is 2 × 2 × 5 × 5 × 5 .
So, 500 is not a perfect cube.
Example 1: Is 243 a perfect cube?
Solution: 243 = 3 × 3 × 3 × 3 × 3
In the above factorisation 3 × 3 remains after grouping the 3’s in triplets. Therefore, 243 is
not a perfect cube.
Which of the following are perfect cubes?
1. 400 2. 3375 3. 8000 4. 15625
5. 9000 6. 6859 7. 2025 8. 10648
6.2.2  Smallest multiple that is a perfect cube
Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm,
30 cm, 15 cm respectively .
Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?
Raj said, V olume of cuboid is 15 × 30 × 15  = 3 × 5 × 2 × 3 × 5 × 3 × 5
  = 2 × 3 × 3 × 3 × 5 × 5 × 5
Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make
it a perfect cube. Therefore, we need 4 such cuboids to make a cube.
Example 2: Is 392 a perfect cube? If not, find the smallest natural number by which
392 must be multiplied so that the product is a perfect cube.
Solution: 392 = 2 × 2 × 2 × 7 × 7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect
cube. To make its a cube, we need one more 7. In that case
392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744           which is a perfect cube.
factors can be
grouped in triples
There are three
5’s in the product but
only two 2’s.
Do you remember that
a
m
 × b
m
 = (a × b)
m
2 216
2 108
2 54
3 27
39
33
1
Reprint 2024-25
Page 5


? ? ? ? ? ? ????? ? ? ???? ? ? ? ? ? ? ?
6.1  Introduction
This is a story about one of  India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
interesting. He said it is the smallest  number that can be expressed
as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12
3
 + 1
3
1729 = 1000 + 729 = 10
3
 + 9
3
1729 has since been known as the Hardy – Ramanujan Number,
even though this feature of 1729 was known more than 300 years
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
6.2  Cubes
You know that the word ‘cube’ is used in geometry . A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
taking it three times.
?????? ???? ????? ?????
? ? ? ? ? ? ?
?
Hardy – Ramanujan
Number
1729 is the smallest Hardy–
Ramanujan Number. There
are an infinitely many such
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
2, 24), Check it with the
numbers given in the brackets.
Figures which have
3-dimensions are known as
solid figures.
Reprint 2024-25
??? ? ? ? ????? ????
The numbers 729, 1000, 1728
are also perfect cubes.
We note that 1 = 1 × 1 × 1 = 1
3
;  8 = 2 × 2 × 2 = 2
3
; 27 = 3 × 3 × 3 = 3
3
.
Since 5
3
 = 5 × 5 × 5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9 = 3 × 3 and there is no natural number which multiplied
by taking three times gives 9. We can see also that 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27. This
shows that 9 is not a perfect cube.
The following are the cubes of numbers from 1 to 10.
Table 1
Number Cube
1 1
3
 = 1
2 2
3
 = 8
3 3
3
 = 27
4 4
3 
= 64
 5 5
3
 = ____
6 6
3
 = ____
7 7
3
 = ____
8 8
3
 = ____
9 9
3
 = ____
10 10
3
 = ____
There are only ten perfect cubes from 1 to 1000. (Check this). How many perfect
cubes are there from 1 to 100?
Observe the cubes of even numbers. Are they all even? What can you say about the
cubes of odd numbers?
Following are the cubes of the numbers from 1 1 to 20.
Table 2
Number Cube
11 1331
12 1728
13 2197
14 2744
15 3375
16 4096
17 4913
18 5832
19 6859
20 8000
We are odd so are
our cubes
We are even, so
are our cubes
Complete it.
Reprint 2024-25
? ? ? ? ? ? ?????? ? ? ??? ? ? ? ? ? ? ? ?
TRY THESE
each prime factor
appears three times
in its cubes
TRY THESE
Consider a few numbers having 1 as the one’ s digit (or unit’ s). Find the cube of each
of them. What can you say about the one’s digit of the cube of a number having 1 as the
one’s digit?
Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc.
Find the one’s digit of the cube of each of the following numbers.
(i) 3331 (ii) 8888 (iii) 149 (iv) 1005
(v) 1024 (vi) 77 (vii) 5022 (viii) 53
6.2.1  Some interesting patterns
1. Adding consecutive odd numbers
Observe the following pattern of sums of odd numbers.
1 = 1= 1
3
3+ 5= 8 = 2
3
7+ 9+ 11 = 27 = 3
3
13 + 15 + 17 + 19 = 64 = 4
3
21 + 23 + 25 + 27 + 29 = 125 = 5
3
Is it not interesting? How many consecutive odd numbers will be needed to obtain
the sum as 10
3
?
Express the following numbers as the sum of odd numbers using the above pattern?
(a) 6
3
(b) 8
3
(c) 7
3
Consider the following pattern.
2
3
 – 1
3
 = 1 + 2 × 1 × 3
3
3
 – 2
3
 = 1 + 3 × 2 × 3
4
3
 – 3
3
 = 1 + 4 × 3 × 3
Using the above pattern, find the value of the following.
(i) 7
3
 – 6
3
(ii) 12
3
 – 11
3
(iii) 20
3
 – 19
3
(iv) 51
3
 – 50
3
2. Cubes and their prime factors
Consider the following prime factorisation of the numbers and their cubes.
Prime factorisation Prime factorisation
of a number of its cube
4 = 2 × 2 4
3
 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
3
 × 2
3
6 = 2 × 3 6
3
 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2
3
 × 3
3
15 = 3 × 5 15
3
 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3
3
 × 5
3
12 = 2 × 2 × 3 12
3
 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
=2
3
 × 2
3
 × 3
3
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
Observe that each prime factor of a number appears
three times in the prime factorisation of its cube.
In the prime factorisation of any number , if each factor
appears three times, then, is the number a perfect cube?
Think about it. Is 216 a perfect cube?
By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3
Each factor appears 3 times. 216 = 2
3
 × 3
3
 = (2 × 3)
3
= 6
3
   which is a perfect cube!
Is 729 a perfect cube? 729 = 3 × 3 × 3 × 3 × 3 × 3
Yes, 729 is a perfect cube.
Now let us check for 500.
Prime factorisation of 500 is 2 × 2 × 5 × 5 × 5 .
So, 500 is not a perfect cube.
Example 1: Is 243 a perfect cube?
Solution: 243 = 3 × 3 × 3 × 3 × 3
In the above factorisation 3 × 3 remains after grouping the 3’s in triplets. Therefore, 243 is
not a perfect cube.
Which of the following are perfect cubes?
1. 400 2. 3375 3. 8000 4. 15625
5. 9000 6. 6859 7. 2025 8. 10648
6.2.2  Smallest multiple that is a perfect cube
Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm,
30 cm, 15 cm respectively .
Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?
Raj said, V olume of cuboid is 15 × 30 × 15  = 3 × 5 × 2 × 3 × 5 × 3 × 5
  = 2 × 3 × 3 × 3 × 5 × 5 × 5
Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make
it a perfect cube. Therefore, we need 4 such cuboids to make a cube.
Example 2: Is 392 a perfect cube? If not, find the smallest natural number by which
392 must be multiplied so that the product is a perfect cube.
Solution: 392 = 2 × 2 × 2 × 7 × 7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect
cube. To make its a cube, we need one more 7. In that case
392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744           which is a perfect cube.
factors can be
grouped in triples
There are three
5’s in the product but
only two 2’s.
Do you remember that
a
m
 × b
m
 = (a × b)
m
2 216
2 108
2 54
3 27
39
33
1
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THINK, DISCUSS AND WRITE
Hence the smallest natural number by which 392 should be multiplied to make a perfect
cube is 7.
Example 3: Is 53240 a perfect cube? If not, then by which smallest natural number
should 53240 be divided so that the quotient is a perfect cube?
Solution: 53240 = 2 × 2 × 2 × 11 × 11 × 11 × 5
The prime factor 5 does not appear in a group of three. So, 53240 is not a perfect cube.
In the factorisation 5 appears only one time. If we divide the number by 5, then the prime
factorisation of the quotient will not contain 5.
So, 53240 ÷ 5 = 2 × 2 × 2 × 11 × 11 × 11
Hence the smallest number by which 53240 should be divided to make it a perfect
cube is 5.
The perfect cube in that case is = 10648.
Example 4: Is 1188 a perfect cube? If not, by which smallest natural number should
1188 be divided so that the quotient is a perfect cube?
Solution: 1188 = 2 × 2 × 3 × 3 × 3 × 11
The primes 2 and 11 do not appear in groups of three. So, 1188 is not a perfect cube. In
the factorisation of 1188 the prime 2 appears only two times and the prime 11 appears
once. So, if we divide 1188 by 2 × 2 × 11 = 44, then the prime factorisation of the
quotient will not contain 2 and 1 1.
Hence the smallest natural number by which 1188 should be divided to make it a
perfect cube is 44.
And the resulting perfect cube is 1188 ÷ 44 = 27 (=3
3
).
Example 5: Is 68600 a perfect cube? If not, find the smallest number by which 68600
must be multiplied to get a perfect cube.
Solution: We have, 68600 = 2 × 2 × 2 × 5 × 5 × 7 × 7 × 7. In this factorisation, we
find that there is no triplet of 5.
So, 68600 is not a perfect cube. To make it  a perfect cube we multiply it by 5.
Thus, 68600 × 5 = 2 × 2 × 2 × 5 × 5 × 5 × 7 × 7 × 7
= 343000,   which is a perfect cube.
Observe that 343 is a perfect cube. From Example 5 we know that 343000 is also
perfect cube.
Check which of the following are perfect cubes. (i) 2700 (ii) 16000 (iii) 64000
(iv) 900 (v) 125000 (vi) 36000 (vii) 21600 (viii) 10,000 (ix) 27000000 (x) 1000.
What pattern do you observe in these perfect cubes?
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