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BOOK 8 ( CBSE ) / Chap7/ Cube And Cube Roots
Chapter Facts.
1. If n is a non zero integer ,then n x n x n written as n3 is called cube of n. Hence cube
of a number can be found by multiplying it by itself two times.
2. Table of cubes up to 20.
x
x3
1
1
2
8
3
27
4
64
5
125
6
216
7
343
8
512
9
729
x
11
12
13
14
15
16
17
18
19
3
x
1331 1728 2197 2744 3385 4096 4913 5832 6859
3. A relationship exists between the units digit of a number and that of its cube.
10
1000
20
8000
If units digit of a number is 1 ,units digit of its cube is 1.
If units digit of a number is 2 ,units digit of its cube is 8.
If units digit of a number is 3 ,units digit of its cube is 7.
If units digit of a number is 4 ,units digit of its cube is 4.
If units digit of a number is 5 ,units digit of its cube is 5.
If units digit of a number is 6 ,units digit of its cube is 6.
If units digit of a number is 7 ,units digit of its cube is 3.
If units digit of a number is 8 ,units digit of its cube is 2.
If units digit of a number is 9 ,units digit of its cube is 9.
If units digit of a number is 0 ,units digit of its cube is 0.
4. Cube of an even number is even and that of an odd number is odd.
5. A number n is called a perfect cube or perfect third power if there is an integer k
such that n = k x k x k. A perfect cube may be positive or negative.
6. If a number ends in zeros then by observation it is possible to say it may be a perfect
cube or not. If the number of zeros in the end is a multiple of three ( three or six or nine
or …) then the number may be a perfect cube . However if the number of zeros in the end
is not a multiple of three then it the number is certainly not a perfect cube.
7. If n is a perfect cube then there must be an integer m such that n = m x m x m. This
integer m is called cube root of n. We write m =
3
n .( read :– m equal to cube root of
n ).
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8. Table of cube roots.
n
3
1
1
n
8
2
27
3
64
4
125
5
216
6
343
7
512
8
729
9
1000
10
n
1331 1728 2197 2744 3385 4096 4913 5832 6859 8000
11
12
13
14
15
16
17
18
19
20
n
9. A relationship exists between the units digit of a perfect cube and that of its cube root.
3

If a units digit of a perfect cube is 1, units digit of its cube root is 1 .

If a units digit of a perfect cube is 2, units digit of its cube root is 8 .

If a units digit of a perfect cube is 3, units digit of its cube root is 7 .

If a units digit of a perfect cube is 4, units digit of its cube root is 4 .

If a units digit of a perfect cube is 5, units digit of its cube root is 5 .

If a units digit of a perfect cube is 6, units digit of its cube root is 6 .

If a units digit of a perfect cube is 7, units digit of its cube root is 3 .

If a units digit of a perfect cube is 8, units digit of its cube root is 2 .

If a units digit of a perfect cube is 9, units digit of its cube root is 9 .

If a units digit of a perfect cube is 0, units digit of its cube root is 0 .
10. If a and b are perfect cubes then
i)
3
a b 3 a 3 b .( this is useful in finding cube roots of large numbers)
ii) for b ≠ 0
3
a 3 a

.
b 3b
11. Cube root of a negative number can be using the following result..
If x is a perfect cube then
3
 x  3 x .
Formative Assessment
A ) Multiple Choice Questions
Tick () the correct option for each of the following :
1. The cube of a number
(a) can be positive or negative
(b) can never be zero
(c) is always positive
(d) is always negative
2. If a number ends in 1, then its cube ends in
(a) 1
(b) 9
(c) 3
(d) 1 or 9
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3. The smallest Hardy – Ramanujan number is
(a) 1728
(b) 1729
(c) 2728
(d) 2729
(c) 729
(d) 21600
4. Which of the following is not a perfect cube?
(a) 1331
(b) 10648
5. The ones digit of the cube root of a perfect cube which has 2 in its ones place is
(a) 2
(b) 8
(c) 2 or 8
(d) 2 or 4
6. If square of a number ends with 1, then its cube ends with
(a) 1 only
(b) 9 only
(c) 1 or 9
(d) 3
(c) 36 m
(d) 28 m
(c) 4.096 m3
(d) 5.12 m3
7. The side of a cube whose volume is 17576 m3 is
(a) 26 m
(b) 24 m
8. The volume of the cube of edge 1.6 m is
(a) 2.56 m3
(b) 40.96 m3
B ) Fill in the blanks:
1. If n is a non zero integer ,then n x n x n written as _____ is called ________ of n.
2. Cube of 13 is _________ .
3. A perfect cube can also be called a perfect _______________________ .
4. If ‘a’ and ‘b’ are perfect cubes ,
3
a b 3
5. If ‘a’ and ‘b’ are perfect cubes and b≠ 0 ,
3
3
a 3

b 3
C) True / False
Say true or false for each of the following.
1. If ones digit of a perfect cube number is 2 ,units digit of its cube is 8.
T/F .
2. A perfect cube cannot be negative.
T/F
3. If ‘n’ is a natural number and cube of n is less than 1000, n = 1,2,3,….., 9.
T/F
4. If a perfect cube ends in 4 , its cube root also ends in 4.
T/F
5. If a number ends in two zeros, its cube will end in six zeros.
T/F
D ) True / False with Justification :
Say true or false for each of the following. Justify your answer ?
1 Cube of an even number is odd and that of an odd number is even.
T/F
2. A perfect cube ends in three zeros. Its cube root will be a multiple of 10 but not of
100.
T/F
E) Matching
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Match the items in column I with those in column II.
Column I ( perfect cube )
(i)
2197
(ii)
185193
(iii) 9261
(iv)
3375
(v)
24389
(vi)
32768
(vii) 97336
(viii 314432
Column II ( units digit of its cube root )
A 3
B 6
C 8
D 1
E 5
F
9
G 2
H 7
)
(ix)
I
39304
4
Summative Assessment
A ) One mark questions
1. What will be the units digit of the cube of each of the following numbers :
a) 121 b) 532.
2. By observing the number of zeros in the end say which of the numbers cannot be perfect cubes
( say yes or no ) a) 729000000 b) 133100000
3. Side of a cube is 11 cm. Find its volume.
B ) Two mark questions
Find the cube root of the following by prime factorization ( 1 – 3 )
1.
343
2. 9261
3.
59319
4. Find the cube roots of 11052 using the given fact: 110592 = 512  216.
Find the cube root of the following numbers ( 5 – 6 )
5.
125
27
 1000
343
6.
C ) Three mark questions
1.
Find the smallest number by which 3087 be multiplied to make it a perfect cube :
2.
Find the smallest number by which 32928 be divided to make it a perfect cube :
3.
Find the cube root of -13824
4. The volume of a box in the shape of a cube is 2.197 cubic metres. Find the length of the edge
of the box.
ANSWERS
Formative
MCQ’s
1
a
2
a
3
b
4
d
5
b
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6
c
7
Fill in the blanks
a
8
1
n3 ; cube
3
third power
5
a,b
True False
1
T
2
F
True False with Justification.
1
2
3
c
2
4
2197
a,b
T
4
T
5
T
F ; Cube of 1 is 1 which is odd. Also cube of 2 is 8 which is an even number
T ; The perfect cube number ends in three zeros so its cube root will end in one
zero. This makes it a multiple of 10. Also it cannot end in two zeros as the
perfect cube number does not end in six zeros. So it wont be a multiple of 100.
Matching
(i) – I, (ii) – H , (iii) –D , (iv) – E , (v) –F, (vi) – G , (vii) -B , (viii) - C , (ix) - A
Summative
One mark questions
1
a) 1 b) 8
3
1331 cu. cm.
Two marks questions
2
a) yes b) no
1
7
3
39
5
5/3
Three marks questions
2
4
6
21
48
- 10/7
1
3
3
- 24
end …
2
4
3
1.3 m
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