*Q1 : *

*By the method of prime factorization find the cube root for the following. *

*(A) 64 *

*(B) 512 *

*(C) 10648*

*(D) 27000*

*(E) 15625 *

*(F) 13824 *

*(G) 110592 *

*(H) 46656*

*(I) 175616*

*(J) 91125*

**Solution:**

(A) 64

= 4

02 | 064 |

02 | 032 |

02 | 016 |

02 | 08 |

02 | 04 |

02 | 02 |

01 |

(B) 512

= 2 x 2 x 2

= 8

02 | 0512 |

02 | 0256 |

02 | 0128 |

02 | 064 |

02 | 032 |

02 | 016 |

02 | 08 |

02 | 04 |

02 | 02 |

01 |

(C) 10648

\sqrt[3]{10648}=\sqrt[3]{2\times 2\times 2\times 11\times 11\times 11}

= 2 x 11

=22

02 | 010648 |

02 | 05324 |

02 | 02662 |

011 | 01331 |

011 | 0121 |

011 | 011 |

01 |

(D) 27000

=>2 x 3 x 5

=>30

02 | 027000 |

02 | 013500 |

02 | 06750 |

03 | 03375 |

03 | 01125 |

03 | 0375 |

05 | 0125 |

05 | 025 |

05 | 05 |

01 |

(E) 15625

=> 5 x 5

=> 25

05 | 015625 |

05 | 03125 |

05 | 0625 |

05 | 0125 |

05 | 025 |

05 | 05 |

01 |

(F) 13824

=> 2 x 2 x 2 x 3

=> 24

02 | 13824 |

02 | 06912 |

02 | 03456 |

02 | 01728 |

02 | 0864 |

02 | 0432 |

02 | 0216 |

02 | 0108 |

02 | 054 |

03 | 27 |

03 | 09 |

03 | 03 |

01 |

(G) 110592

=> 2 x 2 x 2 x 2 x 3

=> 48

02 | 0110592 |

02 | 055296 |

02 | 027648 |

02 | 013824 |

02 | 06912 |

02 | 03456 |

02 | 01728 |

02 | 0864 |

02 | 0432 |

02 | 0216 |

02 | 0108 |

02 | 054 |

03 | 027 |

03 | 09 |

03 | 03 |

01 |

(H) 46656

=> 2 x 2 x 2 x 3 x 3 x 3

=> 36

(I) 175616

=> 2 x 2 x 2 x 7

=> 56

02 | 0175616 |

02 | 087808 |

02 | 043904 |

02 | 021952 |

02 | 010976 |

02 | 05488 |

02 | 02744 |

02 | 01372 |

02 | 0686 |

07 | 0343 |

07 | 049 |

07 | 07 |

01 |

(J) 91125

=> 3 x 3 x 5

=> 45

03 | 091125 |

03 | 010125 |

03 | 03375 |

03 | 01125 |

03 | 0375 |

05 | 0125 |

05 | 025 |

05 | 05 |

01 |

*Q2: *

*State whether the following is true of false:*

*(A) Any off number of a cube is even.*

*(B) When a number end with two zeros, it is never a perfect cube. *

*(C) If the square of a given number ends with 5 then its cube will end with 25. *

*(D) There is no number that ends with 8 which is a perfect cube.*

*(E) The cube of a given two digit number will always be a three digit number. *

*(F) The cube of a two digit number will have either seven or more digits.*

*(G) The cube of single digit number may also be a single digit number.*

**Solution:**

(A) The statement given is false.

Since, 1^{3 } = 1, 3^{3} = 27, 5^{3 }= 125, . . . . . . . . . . are all odd.

(B) The given statement is true.

Since, a perfect cube ends with three zeroes.

Eg. 10^{3} = 1000, 20^{3 }= 8000, 30^{3 } = 27,000 , . . . . . . . . . . . . so on.

(C) The given statement is false

Since, 5^{2} = 25, 5^{3} = 125 , 15^{2 }= 225, 15^{3 }= 3375 ( Did not end with 25)

(D) the given statement is false.

Since 12^{3} = 1728 [the number ends with 8]

22^{3} = 10648 [ the number ends with 8]

(E) The given statement is false

Since, 10^{3} = 1000 [Four digit number]

And 11^{3}= 1331 [four digit number]

(F) The statement is False.

Since 99^{3} = 970299 [Six digit number]

(G) the given statement is true

1^{3} = 1 [single digit]

2^{3 }= 8 [single digit]

*Q3 :*

*1331 is told to be a perfect cube. What are the factorization methods in which you can find its cube root? Similarly, find the cube roots for *

*(i)4913*

*(ii)12167*

*(iii)32768.*

**Solution:**

We know that 10^{3 }= 1000 and possible cute of 11^{3 } = 1331

Since, the cube of units digit is 1^{3 } = 1

Then, cube root of 1331 is 11

(i) 4913

We know that 7^{3} is 343

Next number that comes with 7 as the units place is 17^{3 } = 4913

Therefore the cube root of 4913 is 17

(ii) 12167

Since we know that 3^{3} = 27

Here in cube, the ones digit is 7

Now the next number with 3 In the ones digit is 13^{3} = 2197

And the next number with 3 in the ones digit is 23^{3 } = 12167

Hence the cube root of 12167 is 23

(iii) 32768

We know that 2^{3 }= 8

Here in the cube, the ones digit is 8

Now the next number with 2 in the ones digit is 12^{3 } = 1728

And the next number with 2 as the ones digit 22^{3 }= 10648

And the next number with 2 as the ones digit 32^{3} = 32768

Hence the cube root of 32768 is 32