Title 07 Cubes and Cube Roots(1).pdf ✅

CUBES AND CUBE ROOTS CONTENTS • Cube • Some Interesting Patterns • Cube Root • Digits in cube root of a Number • Sum of Numbers ➢ CUBES A cube is a solid figure which has all its sides equal. If side of a cube is 1 cm then 27 such cubes can make a big cube of side 3 cm. So, no. 1, 8, 27, 64, .... are called perfect cube numbers. Table–1 Number Cube 1 1 3 = 1 2 2 3 = 8 3 3 3 = 27 4 4 3 = 64 5 5 3 = 125 6 6 3 = 216 7 7 3 = 343 8 8 3 = 512 9 9 3 = 729 10 103 = 1000 There are only ten perfect cubes from 1 to 1000 and four from 1 to 100 . Following are the cubes of the numbers from 11 to 20. Table–2 Number Cube 11 1331 12 1728 13 2197 14 2744 15 3375 16 4096 17 4913 18 5832 19 6859 20 8000 Results : 1. Cube of even number is also an even number. 2. Cube of an odd number is also an odd number. 3. Unit place of cube of a number whose unit digit is 2, 3, 7, 8 is 8, 7, 3, 2 respectively ➢ SOME INTERESTING PATTERNS 1. Adding consecutive odd numbers : Observe the following pattern of sums of odd numbers. 1 = 1 = 1 3 3 + 5 = 8 = 2 3 7 + 9 + 11 = 27 = 3 3 13 + 15 + 17 + 19 = 64 = 4 3 21 + 23 + 25 + 27 + 29 = 125 = 5 3 Ex.1 How many consecutive odd numbers will be needed to obtain the sum as 103 ? Sol. 10 (91, 93, 95, 97, 99, 101, 103, 105, 107, 109) 7 CHAPTER
2. Prime factors of perfect cube : Each prime number appears three or multiple of 3 times in its cube. Eg. 8 = 2 × 2 × 2 Eg. 64 = (2 × 2) × (2 × 2) × (2 × 2) Eg 125 = (5 × 5 ×5) = 53 = perfect cube number  a 3 is a perfect cube number. Ex.2 Is 128 a perfect cube ? Sol. 128 = (2 × 2) × (2 × 2) × (2 × 2) × 2 = 27  power of 2 is not a multiple of 3.  it is not a perfect cube. Ex.3 Find the cubes of the following numbers: (a) 2 (b) 3 (c) 7 (d) 0.9 (e) (–5) (f) – 0.1 Sol.(a) 2 × 2 × 2 = 8  2 3 = 8 (b) 3 × 3 × 3 = 27  3 3 = 27 (c) 7 × 7 × 7 = 343  7 3 = 343 (d) 0.9 × 0.9 × 0.9 = 0.729  (0.9) 3 = 0.729 (e) (–5) × (–5) × (–5) = –125 (–5)3 = –125 (f) (–0.1) × (–0.1) × (–0.1) = –0.001  (–0.1)3 = –0.001 A natural number is said to be a perfect cube if it is the cube of another natural number. We know that when odd number of negative factors are multiplied, the product is always negative, so cube can be negative also. ➢ CUBE ROOT If 22 = 4, then the square root of 4, i.e., 4 = 2. Similarly, if 23 = 8, then the cube root of 8 is 2. It is written as 3 8 = 2. If 33 = 27, then the cube root of 27 is 3. Thus, 3 27 = 3. Note that the symbol implied square root. For our convenience, we omit 2 from 2 . But for a cube root, we should use the symbol 3 , and it cannot be omitted also we can use ( )1/3 for cube root. Prime Factorisation Method for Finding the Cube Root Let us take some examples here Ex.4 Find the cube root of 1728. Sol. 3 1728 = (1728)1/3 2 1728 2 864 2 432 2 216 2 108 2 54 3 27 9 3 3 Step : 1 First factorise the given number into its prime factors 3 1728 = 3 2 2 2 2 2 2333 Step : 2 Then group the factors in 3s. 3 1728 = 3 3 3 3 2  2 3 Step : 3 Take one prime factor from each group. 3 1728 = 2 × 2 × 3 = 12  3 1728 = 12 Ex.5 Find the value of 3 216 Sol. 3 216 = (216)1/3 2 216 2 108 2 54 3 27 3 9 3 Step-1: Factorise the given number into its prime factors. 3 3 216 = 2 22333 Step-2: Group the factors in 3s. 3 3 3 3 216 = 2 3
Step-3 : Take one prime factor from each group. 3 216 = 2 × 3 = 6  3 216 = 2 × 3 = 6 Observe 23 = 8, 33 = 27, 43 = 64, 53 = 125,... All cubes of even numbers are even and cubes of odd numbers are odd. Cubes of negative numbers are negative. Ex.6 Find the cube root of 46656. Sol.(i) The unit digit of the number is 6, so the cube root will also have 6 in the unit digit. (ii) Separate the number as 46 656. 46 is greater than 3 3 but less than 43 , so the tens digit is 3. (iii) The required number is 36. Ex.7 Find the cube root of 195112. Sol.(i) Unit digit of the given number is 2, so the required number has unit digit 8. (ii) 195 112, so 195 > 53 but < 63 . So, required number is 58. Note : Above method works for perfect cube numbers called cube root by approximation. ➢ DIGITS IN CUBE ROOT OF A NUMBER Use dots on digit of given number starting from unit digit & leaving 2 next digits, now digits in cube root is same as the sum of dots. Ex.8 Find the digits in cube root of the following numbers. (i) 1728 (ii) 175616 (iii) 8 (iv) 97336 (v) 9261 (vi) 68921000 Sol. • • 1728 two dots  2 digits in cube root • • 175616 two dots  2 digits in cube root • 8 Only one dot  1 digit in cube root • • 9261 Two dots  2 digit in cube root • • • 68921000 Three dots  3 digit in cube root ➢ SUM OF NUMBERS The sum of first ‘n’ natural numbers. 1 + 2 + 3 + ....... + n = 2 n(n +1) Ex.9 Find sum of first 6 natural numbers. Sol. n = 6  Sum = 2 6(6 +1) = 3 × 7 = 21 Ex.10 Find sum of 10 + 11 + ....... + 20. Sol.  Sum of 1 to 20 is 2 20(20 +1) = 10 × 21 = 210 and sum of 1 to 9 is 2 9(9 +1) = 2 910 = 45  10 + 11 + ....... + 20 = 210 – 45 = 165 The sum of Square of first ‘n’ natural numbers. 1 2 + 22 + 32 + .......... + n2 = 6 n(n +1)(2n +1) Ex.11 Find sum of squares of first five natural numbers. Sol. 1 2 + 22 + 32 + 42 + 52  n = 5  sum = 6 5(5 +1)(10 +1) = 55 The sum of cube of first ‘n’ natural numbers. 1 3 + 23 + 33 + 43 + ............ + n 3 = 2 2 n(n 1)       + Ex.12 Find sum of cube of first five natural numbers. Sol. 1 3 + 23 + .......... + 53  n = 5  sum = 2 2 5(5 1)       + = (5 × 3)2 = 225
EXERCISE Q.1 Find the number of digits in the cube root if number of digits in perfect cube numbers as follows. (i) 6 (ii) 5 (iii) 4 (iv) 3 (v) 2 (vi) 1 (vii) 7 Q.2 Find the value of 3 117 + 19683 . Q.3 Which of the following are perfect cube ? (i) 10 (ii) 100 (iii) 1000 (iv) 104 (v) 105 (vi) 106 Q.4 Find the value of 1016064 (2) (10) 3 3 + Q.5 Find the sum of cubes of first 10 natural numbers. Q.6 Find the value of (13 + 23 + 33 + 44 + ....+ 153 ) – (12 + 22 + 32 + .....+ 102 ) Q.7 Find the cube root of the following numbers by inspection. (i) 12167 (ii) 46.656 (iii) 6859 (iv) 912673 (v) 29791 Q.8 Find cube root of [ 5 100 + 49 + (79507)1/3]. Q.9 Find cube root by prime factorisation (i) 4913 (ii) 13824 (iii) 175616 (iv) 456533 Q.10 Find the least number by which when multiply the following numbers, such that the number become perfect cube. (i) 2048 (ii) 1029 (iii) 45 (iv) 5832 Q.11 Find the least number by which when divide the following numbers. The number become perfect cube also find cube root of new number (i) 4394 (ii) 8575 (iii) 7986 (iv) 28672 ANSWER KEY 1. (i) 2 ; (ii) 2 ; (iii) 2 ; (iv) 1 ; (v) 1 ; (vi) 1; (vii) 3 2. 117 + 27 = 12 3. (iii), (vi) 4. 1 5. 3025 6. 14015 7. (i) 23 ; (ii) 3.6; (iii) 19; (iv) 97; (v) 31 8. 10 9. (i) 17;(ii)24; (iii) 56; (iv) 77 10. (i) 2; (ii) 9; (iii) 75; (iv) 1 11. (i) 2, 13; (ii) 25, 7; (iii) 6, 11; (iv) 7, 16