Exercise 4.2 - Cubes & Cube Roots RD Sharma Solutions | Mathematics (Maths) Class 8 PDF Download

 Page 1


       
          
                              
 
       
          
                              
 
                
 
         
 
          
 
          
 
    
 
   
       
   
 
        
              
            
 
   
 
     
Question:24
Find the cubes of:
i -11
ii -12
iii -21
Solution:
i
Cube of -11 is given as:
(-11)
3
= -11 × -11 × -11 = -1331
Thus, the cube of 11 is (-1331).
ii
Cube of -12 is given as:  
(-12)
3
= -12 × -12 × -12 = -1728
Thus, the cube of -12 is (-1728).
iii
Cube of -21 is given as:  
(-21)
3
= -21 × -21 × -21 = -9261
Thus, the cube of -21 is (-9261).
Question:25
Which of the following numbers are cubes of negative integers
i -64
ii -1056
iii -2197
iv -2744
v -42875
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 64 into prime factors, we get:
64 = 2 ×2 ×2 ×2 ×2 ×2
On grouping the factors in triples of equal factors, we get:
Page 2


       
          
                              
 
       
          
                              
 
                
 
         
 
          
 
          
 
    
 
   
       
   
 
        
              
            
 
   
 
     
Question:24
Find the cubes of:
i -11
ii -12
iii -21
Solution:
i
Cube of -11 is given as:
(-11)
3
= -11 × -11 × -11 = -1331
Thus, the cube of 11 is (-1331).
ii
Cube of -12 is given as:  
(-12)
3
= -12 × -12 × -12 = -1728
Thus, the cube of -12 is (-1728).
iii
Cube of -21 is given as:  
(-21)
3
= -21 × -21 × -21 = -9261
Thus, the cube of -21 is (-9261).
Question:25
Which of the following numbers are cubes of negative integers
i -64
ii -1056
iii -2197
iv -2744
v -42875
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 64 into prime factors, we get:
64 = 2 ×2 ×2 ×2 ×2 ×2
On grouping the factors in triples of equal factors, we get:
64 = {2 ×2 ×2}×{2 ×2 ×2}
It is evident that the prime factors of 64 can be grouped into triples of equal factors and no factor is left over. Therefore, 64 is a perfect cube. This implies that -64 is also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 = 4 
This implies that 64 is a cube of 4.
Thus, -64 is the cube of -4.
ii
On factorising 1056 into prime factors, we get:
1056 = 2 ×2 ×2 ×2 ×2 ×3 ×11
On grouping the factors in triples of equal factors, we get: ? 1056 = {2 ×2 ×2}×2 ×2 ×3 ×11
It is evident that the prime factors of 1056 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1056 is not a perfect cube. This implies that -1056 is
not a perfect cube as well.
iii
On factorising 2197 into prime factors, we get:
2197 = 13 ×13 ×13
On grouping the factors in triples of equal factors, we get: ? 2197 = {13 ×13 ×13}
It is evident that the prime factors of 2197 can be grouped into triples of equal factors and no factor is left over. Therefore, 2197 is a perfect cube. This implies that -2197 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus, -2197 is the cube of -13.
iv
On factorising 2744 into prime factors, we get:
2744 = 2 ×2 ×2 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744 = {2 ×2 ×2}×{7 ×7 ×7}
It is evident that the prime factors of 2744 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744 is a perfect cube. This implies that -2744 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×7 = 14
This implies that 2744 is a cube of 14.
Thus, -2744 is the cube of -14.
v
On factorising 42875 into prime factors, we get:
42875 = 5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 42875 = {5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 42875 can be grouped into triples of equal factors and no factor is left over. Therefore, 42875 is a perfect cube. This implies that -42875 is also a
perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
5 ×7 = 35
This implies that 42875 is a cube of 35.
Thus, -42875 is the cube of -35.
Question:26
Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer.
i -5832
ii -2744000
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 5832 into prime factors, we get:
5832 = 2 ×2 ×2 ×3 ×3 ×3 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
5832 = {2 ×2 ×2}×{3 ×3 ×3}×{3 ×3 ×3}
It is evident that the prime factors of 5832 can be grouped into triples of equal factors and no factor is left over. Therefore, 5832 is a perfect cube. This implies that -5832 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get:
2 ×3 ×3 = 18 
This implies that 5832 is a cube of 18.
Thus, -5832 is the cube of -18.
ii
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744000 is a perfect cube. This implies that -2744000 is
also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Thus, -2744000 is the cube of -140.
Question:27
Find the cube of:
i 
7
9
ii -
8
11
iii 
12
7
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Question:24
Find the cubes of:
i -11
ii -12
iii -21
Solution:
i
Cube of -11 is given as:
(-11)
3
= -11 × -11 × -11 = -1331
Thus, the cube of 11 is (-1331).
ii
Cube of -12 is given as:  
(-12)
3
= -12 × -12 × -12 = -1728
Thus, the cube of -12 is (-1728).
iii
Cube of -21 is given as:  
(-21)
3
= -21 × -21 × -21 = -9261
Thus, the cube of -21 is (-9261).
Question:25
Which of the following numbers are cubes of negative integers
i -64
ii -1056
iii -2197
iv -2744
v -42875
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 64 into prime factors, we get:
64 = 2 ×2 ×2 ×2 ×2 ×2
On grouping the factors in triples of equal factors, we get:
64 = {2 ×2 ×2}×{2 ×2 ×2}
It is evident that the prime factors of 64 can be grouped into triples of equal factors and no factor is left over. Therefore, 64 is a perfect cube. This implies that -64 is also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 = 4 
This implies that 64 is a cube of 4.
Thus, -64 is the cube of -4.
ii
On factorising 1056 into prime factors, we get:
1056 = 2 ×2 ×2 ×2 ×2 ×3 ×11
On grouping the factors in triples of equal factors, we get: ? 1056 = {2 ×2 ×2}×2 ×2 ×3 ×11
It is evident that the prime factors of 1056 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1056 is not a perfect cube. This implies that -1056 is
not a perfect cube as well.
iii
On factorising 2197 into prime factors, we get:
2197 = 13 ×13 ×13
On grouping the factors in triples of equal factors, we get: ? 2197 = {13 ×13 ×13}
It is evident that the prime factors of 2197 can be grouped into triples of equal factors and no factor is left over. Therefore, 2197 is a perfect cube. This implies that -2197 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus, -2197 is the cube of -13.
iv
On factorising 2744 into prime factors, we get:
2744 = 2 ×2 ×2 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744 = {2 ×2 ×2}×{7 ×7 ×7}
It is evident that the prime factors of 2744 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744 is a perfect cube. This implies that -2744 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×7 = 14
This implies that 2744 is a cube of 14.
Thus, -2744 is the cube of -14.
v
On factorising 42875 into prime factors, we get:
42875 = 5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 42875 = {5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 42875 can be grouped into triples of equal factors and no factor is left over. Therefore, 42875 is a perfect cube. This implies that -42875 is also a
perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
5 ×7 = 35
This implies that 42875 is a cube of 35.
Thus, -42875 is the cube of -35.
Question:26
Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer.
i -5832
ii -2744000
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 5832 into prime factors, we get:
5832 = 2 ×2 ×2 ×3 ×3 ×3 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
5832 = {2 ×2 ×2}×{3 ×3 ×3}×{3 ×3 ×3}
It is evident that the prime factors of 5832 can be grouped into triples of equal factors and no factor is left over. Therefore, 5832 is a perfect cube. This implies that -5832 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get:
2 ×3 ×3 = 18 
This implies that 5832 is a cube of 18.
Thus, -5832 is the cube of -18.
ii
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744000 is a perfect cube. This implies that -2744000 is
also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Thus, -2744000 is the cube of -140.
Question:27
Find the cube of:
i 
7
9
ii -
8
11
iii 
12
7
iv -
13
8
v 2
2
5
vi 3
1
4
vii 0.3
viii 1.5
ix 0.08
x 2.1
Solution:
i
? 
m
n
3
=
m
3
n
3
? 
7
9
3
=
7
3
9
3
=
7×7×7
9×9×9
=
343
729
ii
? -
m
n
3
= -
m
3
n
3
 
? -
8
11
3
= -
8
11
3
= -
8
3
11
3
= -
8×8×8
11×11×11
= -
512
1331
iii
? 
m
n
3
=
m
3
n
3
? 
12
7
3
=
12
3
7
3
=
12×12×12
7×7×7
=
1728
343
iv
? -
m
n
3
= -
m
3
n
3
? -
13
8
3
= -
13
8
3
= -
13
3
8
3
= -
13×13×13
8×8×8
= -
2197
512
v
We have:
2
2
5
=
12
5
Also, 
m
n
3
=
m
3
n
3
? 
12
5
3
=
12
3
5
3
=
12×12×12
5×5×5
=
1728
125
vi
We have:
3
1
4
=
13
4
Also, 
m
n
3
=
m
3
n
3
? 
13
4
3
=
13
3
4
3
=
13×13×13
4×4×4
=
2197
64
vii
We have:
0. 3 =
3
10
Also, 
m
n
3
=
m
3
n
3
? 
3
10
3
=
3
3
10
3
=
3×3×3
10×10×10
=
27
1000
= 0. 027
viii
We have:
1. 5 =
15
10
Also, 
m
n
3
=
m
3
n
3
? 
15
10
3
=
15
3
10
3
=
15×15×15
10×10×10
=
3375
1000
= 3. 375
( )
( )
( )
( ) ( ) ( ) ( )
( )
( )
( )
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
Page 4


       
          
                              
 
       
          
                              
 
                
 
         
 
          
 
          
 
    
 
   
       
   
 
        
              
            
 
   
 
     
Question:24
Find the cubes of:
i -11
ii -12
iii -21
Solution:
i
Cube of -11 is given as:
(-11)
3
= -11 × -11 × -11 = -1331
Thus, the cube of 11 is (-1331).
ii
Cube of -12 is given as:  
(-12)
3
= -12 × -12 × -12 = -1728
Thus, the cube of -12 is (-1728).
iii
Cube of -21 is given as:  
(-21)
3
= -21 × -21 × -21 = -9261
Thus, the cube of -21 is (-9261).
Question:25
Which of the following numbers are cubes of negative integers
i -64
ii -1056
iii -2197
iv -2744
v -42875
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 64 into prime factors, we get:
64 = 2 ×2 ×2 ×2 ×2 ×2
On grouping the factors in triples of equal factors, we get:
64 = {2 ×2 ×2}×{2 ×2 ×2}
It is evident that the prime factors of 64 can be grouped into triples of equal factors and no factor is left over. Therefore, 64 is a perfect cube. This implies that -64 is also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 = 4 
This implies that 64 is a cube of 4.
Thus, -64 is the cube of -4.
ii
On factorising 1056 into prime factors, we get:
1056 = 2 ×2 ×2 ×2 ×2 ×3 ×11
On grouping the factors in triples of equal factors, we get: ? 1056 = {2 ×2 ×2}×2 ×2 ×3 ×11
It is evident that the prime factors of 1056 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1056 is not a perfect cube. This implies that -1056 is
not a perfect cube as well.
iii
On factorising 2197 into prime factors, we get:
2197 = 13 ×13 ×13
On grouping the factors in triples of equal factors, we get: ? 2197 = {13 ×13 ×13}
It is evident that the prime factors of 2197 can be grouped into triples of equal factors and no factor is left over. Therefore, 2197 is a perfect cube. This implies that -2197 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus, -2197 is the cube of -13.
iv
On factorising 2744 into prime factors, we get:
2744 = 2 ×2 ×2 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744 = {2 ×2 ×2}×{7 ×7 ×7}
It is evident that the prime factors of 2744 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744 is a perfect cube. This implies that -2744 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×7 = 14
This implies that 2744 is a cube of 14.
Thus, -2744 is the cube of -14.
v
On factorising 42875 into prime factors, we get:
42875 = 5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 42875 = {5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 42875 can be grouped into triples of equal factors and no factor is left over. Therefore, 42875 is a perfect cube. This implies that -42875 is also a
perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
5 ×7 = 35
This implies that 42875 is a cube of 35.
Thus, -42875 is the cube of -35.
Question:26
Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer.
i -5832
ii -2744000
Solution:
In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m
3
 is the cube of -m.
i
On factorising 5832 into prime factors, we get:
5832 = 2 ×2 ×2 ×3 ×3 ×3 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
5832 = {2 ×2 ×2}×{3 ×3 ×3}×{3 ×3 ×3}
It is evident that the prime factors of 5832 can be grouped into triples of equal factors and no factor is left over. Therefore, 5832 is a perfect cube. This implies that -5832 is also a perfect
cube.
Now, collect one factor from each triplet and multiply, we get:
2 ×3 ×3 = 18 
This implies that 5832 is a cube of 18.
Thus, -5832 is the cube of -18.
ii
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get: ? 2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over. Therefore, 2744000 is a perfect cube. This implies that -2744000 is
also a perfect cube.
Now, collect one factor from each triplet and multiply, we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Thus, -2744000 is the cube of -140.
Question:27
Find the cube of:
i 
7
9
ii -
8
11
iii 
12
7
iv -
13
8
v 2
2
5
vi 3
1
4
vii 0.3
viii 1.5
ix 0.08
x 2.1
Solution:
i
? 
m
n
3
=
m
3
n
3
? 
7
9
3
=
7
3
9
3
=
7×7×7
9×9×9
=
343
729
ii
? -
m
n
3
= -
m
3
n
3
 
? -
8
11
3
= -
8
11
3
= -
8
3
11
3
= -
8×8×8
11×11×11
= -
512
1331
iii
? 
m
n
3
=
m
3
n
3
? 
12
7
3
=
12
3
7
3
=
12×12×12
7×7×7
=
1728
343
iv
? -
m
n
3
= -
m
3
n
3
? -
13
8
3
= -
13
8
3
= -
13
3
8
3
= -
13×13×13
8×8×8
= -
2197
512
v
We have:
2
2
5
=
12
5
Also, 
m
n
3
=
m
3
n
3
? 
12
5
3
=
12
3
5
3
=
12×12×12
5×5×5
=
1728
125
vi
We have:
3
1
4
=
13
4
Also, 
m
n
3
=
m
3
n
3
? 
13
4
3
=
13
3
4
3
=
13×13×13
4×4×4
=
2197
64
vii
We have:
0. 3 =
3
10
Also, 
m
n
3
=
m
3
n
3
? 
3
10
3
=
3
3
10
3
=
3×3×3
10×10×10
=
27
1000
= 0. 027
viii
We have:
1. 5 =
15
10
Also, 
m
n
3
=
m
3
n
3
? 
15
10
3
=
15
3
10
3
=
15×15×15
10×10×10
=
3375
1000
= 3. 375
( )
( )
( )
( ) ( ) ( ) ( )
( )
( )
( )
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
ix
We have:
0. 08 =
8
100
Also, 
m
n
3
=
m
3
n
3
? 
8
100
3
=
8
3
100
3
=
8×8×8
100×100×100
=
512
1000000
= 0. 000512
x
We have:
2. 1 =
21
10
Also, 
m
n
3
=
m
3
n
3
? 
21
10
3
=
21
3
10
3
=
21×21×21
10×10×10
=
9261
1000
= 9. 261
Question:28
Find which of the following numbers are cubes of rational numbers:
i 
27
64
ii 
125
128
iii 0.001331
iv 0.04
Solution:
i
We have:
27
64
=
3×3×3
8×8×8
=
3
3
8
3
=
3
8
3
Therefore, 
27
64
 is a cube of 
3
8
.
ii
We have:
125
128
=
5×5×5
2×2×2×2×2×2×2
=
5
3
2
3
×2
3
×2
It is evident that 128 cannot be grouped into triples of equal factors; therefore, 
125
128
 is not a cube of a rational number.
iii
We have:
0. 001331 =
1331
1000000
=
11×11×11
2×2×2×2×2×2×5×5×5×5×5×5
=
11
3
(2×2×5×5)
3
=
11
3
100
3
=
11
100
3
Therefore, 0. 001331 is a cube of 
11
100
.
iv
We have:
0. 04 =
4
100
=
2×2
2×2×5×5
It is evident that 4 and 100 could not be grouped in to triples of equal factors; therefore, 0.04 is not a cube of a rational number.
            
            
 
 
 
 
                                    
     
 
 
                                                                             
     
 
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( )
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( )
( )