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Q.3 You are told that 1,331 is a perfect cube. Can you guess without factorisation what
is its cube root? Similarly, guess the cube roots of

4913,

12167,

32768.

We have 1331. Divide number in two parts:   First part is 1 and second is 331.   Since the given number is ending with 1 so the last digit of cube root will be 1. In the first part, we have 1. So   By estimation, the cube root of 1331 is 11. Similarly for all other parts. 4913:-    First part is 4 and the second part is 913.               The number is ending with 3 so its cube root will have 7...

2. State true or false.
(i)  Cube of any odd number is even.

(ii)  A perfect cube does not end with two zeros.

(iii)  If square of a number ends with 5, then its cube ends with 25.

(iv)  There is no perfect cube which ends with 8

(v)  The cube of a two digit number may be a three digit number.

(vi)  The cube of a two digit number may have seven or more digits.

(vii)  The cube of a single digit number may be a single digit number.

(i) False.  Cube of an odd number can never be even. (ii) True.  Perfect cube number ends with zeros multiple of three. (iii) False. We can say only about units place. (iv) False. Cube of numbers which ends with 2 end with 8. (v) False. Can never be. (vi) False. Can never be. It can be proved by taking examples. (vii) True. e.g. 1,2

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(x) 91125

The detailed solution for the above-written question is as follows By prime factorization, we get :                   So its cube root is      =   45.

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(ix) 175616

The detailed solution for the above-written question is as follows By prime factorization we get :                       So its cube root is     =  56.

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(viii) 46656

The detailed solution for the above-written question is as follows By prime factorization, we get :                   So its cube root is    =   36.

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(vii) 110592

The detailed solution for the above-written question is as follows By prime factorization:                      So its cube root is  = 48.

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(vi) 13824

The detailed solution for the above-written question is as follows By prime factorization:                  So its cube root is 24.

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(v) 15625

The detailed solution for the above-written question is as follows By prime factorization:       So its cube root is 25.

Q.1. Find the cube root of each of the following numbers by prime factorisation method.

(iv) 27000

The detailed solution for the above-written question is as follows By prime factorization method, we get :        So its cube root is 30.

Q.1  Find the cube root of each of the following numbers by prime factorisation method.

(iii) 10648

The detailed solution for the above-written question is as follows Prime factorization of 10648 gives :             So its cube root is 22.

Q.1  Find the cube root of each of the following numbers by prime factorisation method.

(ii) 512

By prime factorisation of 512 :         So its cube root is

Q.1 Find the cube root of each of the following numbers by prime factorisation method.

(i) 64

The detailed solution for the above-written question is as follows Prime factorization of 64 gives :         So its cube root is    = 4

Q. State true or false:

for any integer $\inline m,m^{2}< m^{3}.$ Why?

The detailed solution for the above-written question is as follows. False.                   or            or           or          Now put any number less than 1, we see that this relation doesn't hold. So for m<1 this condition is not true.

Q.4  Parikshit makes a cuboid of plasticine of sides $\inline 5\: cm,2\; cm,5\; cm$. How many such cuboids will he need to form a cube?

Volume of cuboid is    To make it a cube need to make this a pefect cube number. So we need   cuboids  or      20 cuboids.

Q.3  Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81

(ii) 128

(iii) 135

(iv) 192

(v) 704

By prime factorization of given numbers :  (i) 81 :                     So given number needs to be divided by 3 to get a perfect cube. (ii) 128 :    .               So the given number needs to be divided by 2 to get a perfect cube. (iii) 135 :                   So the given number needs to be divided by 5 to get a perfect cube. (iv) 192 :                   So the given number needs to be...

Q.2 Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243

(ii) 256

(iii) 72

(iv) 675

(v) 100

This can be found by knowing about the prime factors of the number. (i) 243 :     .                 So it must be multiplied by 3. (ii) 256 :                     So the given number must be multiplied by 2 to make it a perfect cube. (iii) 72 :                     So 72 must be multiplied by 3 to make it a perfect cube. (iv) 675 :                   So it should be multiplied by 5. (v) 100 :     ...

Q.1 Which of the following numbers are not perfect cubes?

(v) 46656

We have 46656, by prime factorisation:        . Since prime numbers are in group of three. So the given number is a perfect cube.

Q.1 Which of the following numbers are not perfect cubes?

(iv) 100

The detailed solution for the above-written question is as follows By prime factorization of 100 :      . Since prime numbers are not in pair of three so given number is not a perfect cube.

Q.1 Which of the following numbers are not perfect cubes?

(iii) 1000

The detailed solution for the above written is as follows By prime factorization of 1000 we get :               .  So the given number is a perfect cube.

Q.1 Which of the following numbers are not perfect cubes?

(ii) 128

We have 128.  By prime factorization we get,           Since the prime numbers are not in pairs of three, so the given number is not a perfect cube.
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