CONTENTS Introduction Perfect Cubes Cube root by prime

CONTENTS �Introduction �Perfect Cubes �Cube root by prime factorisation �Cube root by estimation.

INTRODUCTION 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. 3 3 1729 = 1728+1=12 + 1 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 = 13 or 1 cube, 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 Ø Ø 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 ”

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

v EXAMPLE ; Find the cube root of 8000. ANSWER: Prime factorisation of 8000 is 8000=2× 2× 2× 5× 5× 5 Therefore “ ” = 2× 2× 5=20

CUBE ROOT BY ESTIMATION METHOD To find the cube root of a cube number, the following method can be used. Ø STEP 1. 857375= 857 375 ↓ Take a cube number; 857375. second number first number We get 375 & 857 as two groups of Make group of three digits each starting from the right most digit of the number.

STEP 2 375 ‾ So, we get 5 at the unit’s place cube root. Ø First group i. e. , 375 will give the Ø STEP 3 one’s digit of the required cube 857 We know that 9^3=729 & 10^3=1000. Also, root. 729<857<1000. We take the one”s place, place the The number 375 ends with 5. We as the ten “s place of the required cube root. So, we get “CR” 857375=95 know that 5 comes at the unit’s place of a number only when it’s cube root ends in 5.

WORK SHEET (FOR FA-3) Find the cube root of each 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=4 ‾‾‾‾‾ ii. 3√ 512=2× 2× 2 =2× 2× 2=8 ‾‾‾‾‾ iii. 3√ 10648=11× 11× 2× 2× 2=11× 2=22 ‾‾‾‾‾‾‾ i. 26 ii 15 iii 11

1. 2. 3. WHICH OF THE FOLLOWING ARE NOT PERFECT CUBE 216 128 1000 ANS – 128 IS NOT A PERFECT CUBE 1. 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.

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