CUBES ANDCUBE ROOTS

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IntroductionPerfect CubesCube rootCube root by prime factorisation

Cube root by estimation.

CONTENTS

one of the great mathematical geniuses, S ramanujan had a visit of prof G H hardy. He came with a taxi whose number is 1729 .he described the number as a dull number. Ramanujan quickly pointed that 1729 was indeed interesting. He said it was the smallest number that can be expressed as the sum of two cubes in two different ways.

1729 = 1728+1=123 + 13

1729 = 1000+729=103 + 93

INTRODUCTION

A3

WHAT IS A CUBENumbers like 1, 8 ,27… are called cube numbers or a perfect cube . We get perfect cubes by multiplying a number3 times with the same number.

For example- 1×1×1 = 13 or 1cube, 8= 23 , 27= 33…etc. No. Cube.

1 1

2 8

3 27

4 64

5 125

6 216

7 343

8 512

9 729

10 1000

11 1331

12 1728

13 2197

14 2744

15 3375

16 4096

17 4913

18 5332

19 6859

20 8000

CUBE ROOTS

Finding the square root, as you know , is the inverse operation of squaring. Similarly finding the cube root is the inverse operation of finding cube.

We know that 23= 8; so we say that cube root of 8 is 2. We write = 2.

83=512;so of =8. THE SYMBOL “ ” DENOTES “CUBE ROOT

” 3

3 8

3 512

CUBE ROOT THROUGH FACTORISATION METHOD

CONSIDER “3375”: 3375=3×3×3×5×5×5 ‾‾‾‾‾ ‾‾‾‾‾ =33 × 53=(3 × 5)3

= 15

We find its cube root by prime factorisation.

The factors are ; Therefore, cube root of 3375= 15

3 3375

EXAMPLE ; Find the cube root of 8000. ANSWER:

Prime factorisation of 8000 is

8000=2×2×2×2×2×2×5×5×5

Therefore “ ” = 2×2×5=20

3 8000

CUBE ROOT BY ESTIMATIONMETHOD

To find the cube root of a cube number, the following method can be used.

STEP 1.

857375= 857 375 ↓ ↓ second number first number

We get 375 & 857 as two groups of three digits each

Take a cube number; 857375. Make group of three digits starting from the right most digit of the number.

STEP 2 375 ‾ So,we get 5 at the unit’s place cube root.

STEP 3 857 We know that 9^3=729 & 10^3=1000.Also,

729<857<1000.We take the one”s place,place the as the ten “s place of the required cube root.So,we get “CR”857375=95

First group i.e.,375 will give the one’s digit of the required cube root.The number 375 ends with 5.We know that 5 comes at the unit’s place of a number only when it’s cube root ends in 5. Now we take the next group

WORK SHEET (FOR FA-3) Find the cube root of each of the of the following by prime factorization method. 64 512 Find the cube root through estimation i.17576 ii. 3375 iii.1331    ANSWER

i. 3√64=2×2×2×2×2×2=2×2=4 ‾‾‾‾‾ ‾‾‾‾‾ ii.3√512=2×2×2×2×2×2×2×2×2 =2×2×2=8 ‾‾‾‾‾ ‾‾‾‾‾ ‾‾‾‾‾ iii.3√10648=11×11×11×2×2×2=11×2=22 ‾‾‾‾‾‾‾ ‾‾‾‾‾ i. 26 ii 15 iii 11

WHICH OF THE FOLLOWING ARE NOT PERFECT CUBE

1. 216 2. 1283. 1000

ANS – 128 IS NOT A PERFECT CUBE

4. IS 68600 A PERFECT CUBE ? IF NOT FIND THE SMALLEST NUMBER BY WHICH IT SHOULD BE MULTIPLIED TO GET A PERFECT CUBE.

ANS – NO IT’S NOT A PERFECT CUBE. IT SHOULD BE MULTIPLIED BY 5

IS 1188 A PERFECT CUBE ?IF NOT, BY WHICH SMALLEST NATURAL NUMBER SHOULD IT BE DIVIDED SO THAT THE QUOTIENT IS A PERFECT CUBE.

ANS –NO IT’S NOT A PERFECT CUBE , IT SHOULD BE DIVIDED BY 44 TO GET A PERFECT CUBE.

Thank

you