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Question 1

Find the cubes of the following numbers:  (i) 7, (ii) 12, (iii) 21, (iv) 100, (v) 302

Solution:
(i)   (7)3  =  7×7×7 = 343

(ii)   (12)3  = 12 × 12 × 12 = 1728

(iii)   (21)3  = 21 × 21 × 21 = 9621

(iv)   (100)3 = 100 × 100 × 100 = 1000000

(v)   (302)3 = 302 × 302 × 302 = 27543608
 

Question 2

Write cubes of all natural numbers between 1 and 20 and verify the following statements:

       (a) Cubes of all odd natural numbers are odd.
       (b) Cubes of all even natural numbers are even.

Solution:
(2)3 = 8, (3)3= 27, (4)3= 64, (5)3 = 125, (6)3 = 216, ...  (19)3=6859.

(a) Yes, cubes of all odd natural numbers are odd. 
(b) Yes, cubes of all even natural numbers are even.

Question 3

Write cubes of 5 natural numbers which are multiples of 3 and verify the following:
       'The cube of natural number, which is a multiple of 3 is a multiple of 27'.

Solution:
(3)3 = 3× 3× 3 =27
(6)3 = 6
× 6× 6 =216
(9)3 = 9
× 9 × 9 =729
(12)3 = 12
× 12 × 12 =1728
(15)3 = 15
× 15 × 15 =3375

Verification:

(3)3 = 27 = 27
× 1
(6)3 = 216 = 27
× 8
(9)3 = 729 = 27
× 27
(12)3 = 1728 = 27
× 64
(15)3 = 3375 = 27
× 125

... 'The cube of natural number, which is a multiple of 3 is a multiple of 27'.



Solution:
The 5 natural numbers which are of the form 3n + 1 (e.g  4, 7, 10, …) are as follows:
3 ×1 + 1 = 3 + 1 = 4
3
×2 + 1 = 6 + 1 = 7
3
×3 + 1 = 9 + 1 = 10
3
×4 + 1 = 12 + 1 = 13
3
×5 + 1 = 15 + 1 = 16 

The cubes of 5 natural numbers which are of the form 3n + 1 (e.g  4, 7, 10, …) are as follows:

(4)3 = 4
× 4 ×4 = 64
(7)3 = 7
×7 ×7 = 343
(10)3 = 10
×10 ×10 = 1000
(13)3 = 13
×13 ×13 = 2197
(16)3 = 16
× 16 × 16 = 4096 

Verification:
64 = 3 
× 21 + 1
343 = 3
× 114 + 1
1000 = 3
× 333 + 1
2197 = 3
× 732 + 1
4096 = 3
× 1365 + 1
...  'The cube of a natural number of the form 3n +1 is a natural number of the same form'.


Question 5

Write cubes of 5 natural numbers which are of the form 3n + 2 (e.g. 5, 8, 11, …) and verify the following: 'The cube of a natural number of the form 3n + 2 is a natural number of the same form'.

Solution:
The 5 natural numbers which are of the form 3n + 2 (e.g 5, 8, 11 , …) are as follows:
3 ×1 + 2 = 3 + 2 = 5
3
×2 + 2 = 6 + 2 = 8
3
×3 + 2 = 9 + 2 = 11
3
× 4 + 2 = 12 + 2= 14
3
×5 + 2 = 15 + 2 = 17 

The cubes of 5 natural numbers which are of the form 3n + 2 (e.g 5, 8, 11, …) are as follows:

(5)3 = 5
×5 × 5 = 125
(8)3 = 8
× 8 × 8 = 512
(11)3 = 11
× 11 × 11 = 1331
(14)3 = 14
× 14 × 14 = 2744
(17)3 = 17
× 17 × 17 = 4913 
Verification:

125  = 3
× 41 + 2
512  = 3
× 170 + 2
1331 = 3
× 443 + 2
2744 = 3
× 914 + 2
4913 = 3
× 1637 + 2
... 'The cube of a natural number of the form 3n + 2 is a natural number of the same form'.


Question 6

Which of the following numbers are perfect cubes? 1728, 106480 

Solution:



Question 7

What is the smallest number by which 392 must be multiplied so that the product is a perfect cube ?


Solution:
392  = 2 × 2 × 2 × 7 × 7                                                                                       
7 occurs as a prime factor only twice.
Hence, 7 is the smallest number by which 392 must be multiplied so that the product is 
a perfect cube.  


                             

Question 8

What is the smallest number by which 8640 must be divided so that the quotient is a perfect cube ?

Solution:
8640 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5
5 occurs as a prime number only once.
Hence, 5 is the smallest number by which 8640 must be 
divided so that the quotient is a perfect cube.  
                                       



Question 9

If one side of a cube is 13 metres, find its volume.

Solution:
The volume of a cube = (side)3 = (13)3 = 2197m3.

Question 10

Find the cube root of:
         (i) 343     (ii) 1000    (iii)2744    (iv) 74088 

Solution:
(i) 343 = 7 × 7 × 7                                            


...
= 7

(ii) 1000 = 2 × 2 × 2 × 5 × 5 × 5                               


...
= 2 × 5 = 10

(iii) 2744 = 2 × 2 × 2 × 7 × 7 × 7                               


...
= 2 × 7 = 14

(iv) 74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7                  


... = 2 
× × 7 = 42

Question 11

Find the cube root of 125.

Solution:
         

Since we had to subtract five times, therefore,


Question 12

Multiply 137592 by the smallest number so that the product is a perfect cube. Also, find the cube root of the product.

Solution:
137592 = 2´2´2´3´3´3´7´7´13
The number 7 and 13 should be multiplied once and twice
respectively so that the product is a perfect cube.
...  The smallest number by which 137592 must 
be multiplied = 7
´13´13 = 1183 

The required product = 137592 x 1183 =  2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 13 x 7 x 13 x 13
= (23 x 33 x 73 x 133)
= (2 x 3 x 7 x 13)3


                          = 546


Question 13

Divide the number 26244 by the smallest number so that the quotient is a perfect cube. Also, find the cube root of the quotient. 


Solution:


26244 = 2×2×3×3×3×3×3×3×3×3                                  


2x2x 3x 3 = 36 is the smallest number by which 26244 must be divided so that the quotient is a perfect cube. 

       
The required quotient is = 729
The cube root of the quotient = 


Solution:
We know that, the volume of a cube = (side)3
The length of the side of a cube =

Question 15

Which of the following numbers are cubes of negative integers?
         (a) -64      (b) -2197       (c) -1056        (d) -3888

Solution:
(a)    64 = 2x2x2x2x2x2 
     
...     -64 is a cube of -4 a negative integer.

(b)   2197 = 13x13x13
    
...   -2197 is a cube of -13 a negative integer. 

(c)   1056 = 2x2x2x2x2x3x11

In the above factorisation 2 x 3 x 3 x 11 x 11 remains after grouping in triplets. Therefore, 1056 is not a perfect cube.

 Hence -1056 is not a cube of negative integer.

(d)   3888 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 

In the above factorisation 2 x 3 x 3 remains after grouping in triplets. Therefore, 3888 is not a perfect cube.

Hence -3888 is not a cube of negative integer.

Question 16

Find the cube roots of:

         (a) -125       (b) -5832       (c) -17576 

Solution:
(a)     125  =  5x5x5
       

(b)     5832 = 2x2x2x3x3x3x3x3x3
       

(c)     17576  =  2x2x2x13x13x13
       

Question 17

Find the cube root of each of the following numbers:
         1. 8 ×64 
         2. (-216) × 1728
         3. 27 × (-2744)
         4. (-125)×(-3375)
         5. -456533 
         6. -5832000 

Solution:
(1)     8 x 64 = 2x2x2x2x2x2x2x2x2 
     

(2)    216  x 1728 = 2x2x2x3x3x3x2x2x2x2x2x2x3x3x3
      

(3)   27  x  2744 = 3x3x3x2x2x2x7x7x7
     

(4)   125  x 3375 = 5x5x5x3x3x3x5x5x5
     

(5)    456533 = 7x7x7x11x11x11
      

(6)    5832000  =  5832  x 1000 = 2x2x2x3x3x3x3x3x3x2x2x2x5x5x5
      

Question 18

Find the cubes of the following by multiplication.
       (i)                -4
       (ii)              23
       (iii)            3030
 

Solution:
(i)  (-4)3 =(4) x (4) x (4) = 64

(ii) (23)3 =23 x 23 x 23 = 12167

(iii) (3030)3 =3030 x 3030 x 3030 = 27818127000.


Question 19

Find the cube of the following rational numbers:
       (i) 1.4

Solution:

 (i) (1.4)3 = 1.4 × 1.4 × 1.4 = 2.744.

Question 20

 

By what number would you multiply 231525 to make it a perfect cube?

Solution:
The prime factorisation of 231525 is 5 × 5 × 3 × 3 × 3 × 7 × 7 × 7.

The number that must be multiplied in order that the above product is a perfect cube is 5.

Therefore, Cube root of 231525 × 5 is 5 × 3 × 7 = 105.





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