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Cubes and Cube roots

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Cubes and

Cube roots

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Contents

- Introduction
- Perfect Cubes
- Cube root
- Cube root by prime factorisation
- Cube root by estimation.

introduction

- one of the great mathematical geniuses, S ramanujanhad 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

What is a cube

Numbers like 1, 8 ,27… are called cube numbers or a perfect cube . We get perfect cubes by multiplying a number

3 times with the same number.

A

3

- For example- 1×1×1 = 13or 1cube, 8= 23 , 27= 33…etc.

- 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 ”

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

- 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

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”splace,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
- 216
- 128
- 1000
ANS – 128 IS NOT A PERFECT CUBE

- 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