Understanding Cube Roots and Cube Root Calculation Methods

Explore the concept of cube roots and various methods for calculating cube roots such as prime factorization and estimation. Learn how to find the length of the side of a cube given its volume. Dive into examples and see how cube roots are the inverse operation of finding the cube. Gain insights into cube roots of numbers through interesting illustrations and statements.

• Mathematics
• Cube Roots
• Cube Root Calculation
• Prime Factorization
• Estimation

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1. Class VIII Mathematics Chapter 7 Cubes and Cube Roots Module 3 of 3 Prepared by Sumithra Madathil TGT(Maths/Physics) A E C S 2 , Mumbai Atomic Energy Education Society/Distance Learning Programme/2020 1

2. In this Module we will learn about. 1. Cube Roots 2. Cube root of a cube number through prime factorisation method 3.Cube root of a cube number through estimation method Atomic Energy Education Society/Distance Learning Programme/2020 2

3. . If the volume of a cube is 125 cm3, what would be the length of its side? To get the length of the side of the cube, we need to know a number whose cube is 125. Let us explore .. Atomic Energy Education Society/Distance Learning Programme/2020 3

4. . Cube Roots 4 Atomic Energy Education Society/Distance Learning Programme/2020

5. Cube Cube Root Finding the cube root is the inverse operation of finding the cube. Cubed 327 Cube Root ??? = 3) 3 3 = 27 ; the cube root of 27 is 3 ( Atomic Energy Education Society/Distance Learning Programme/2020 5

6. . 23 = 8; so we say that the cube root of 8 is 2; 38 = 2. which is denoted as ? The symbol denotes cube-root. 6 Atomic Energy Education Society/Distance Learning Programme/2020

7. Statement 1 3 = 1 Inference ?? = 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 7 Atomic Energy Education Society/Distance Learning Programme/2020

8. . Cube root through Prime Factorisation method The cube root of a number can be found out by the prime factorisation method. Let us take an example to understand this 8 Atomic Energy Education Society/Distance Learning Programme/2020

9. Let us find the cube root of 3375 by prime factorisation method. First find the prime factorisation of 3375. 3375 = 3 x 3 x 3 x 5 x 5 x 5 3 3375 = 3 3 x 5 3 3 1125 = ( 3 x 5 ) 3 3 375 5 125 Therefore, 5 25 33375 5 5 = 3 x 5 1 = 15 cube root of 3375 = Atomic Energy Education Society/Distance Learning Programme/2020 9

10. Another example Find the cube root of 13824. Step 1 Write 13824 as a product of its prime factors(prime factorisation of 13824) 13824 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 Step 2 Grouping of like prime factors in triplets 13824 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 = 2 3 x 2 3 x 2 3 x 3 3 = ( 2 x 2 x 2 x 3) 3 Step 3 Find cube root 313824 = 3(2 2 2 3)3 = 24 10 Atomic Energy Education Society/Distance Learning Programme/2020

11. More problems 1.Find the cube root of 15625 by prime factorisation method. Solution: The prime factorisation of 15625 is : 15625 = 5 x 5 x 5 x 5 x 5 x 5 = 5 3 x 5 3 = (5 x 5 ) 3 = 25 3 Cube root of 15625 = = 25 (or 315625 = 3 253 315625 = 5 x 5 = 25) 11 Atomic Energy Education Society/Distance Learning Programme/2020

12. . 2. Find the cube root of 27000. Solution: Prime factorisation of 27000 = 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5 = 2 3 x 3 3 x 5 3 327000 = 2 x 3 x 5 Cube root of 27000 = = 30 12 Atomic Energy Education Society/Distance Learning Programme/2020

13. . 3. Find the cube root of 91125. Solution : Prime factorisation of 91125 = 3 x 3 x 3 x 3 x 3 x 3 x 5 x 5 x 5 Cube root of 91125 = = 45 i.e., 391125 = 3 x 3 x 5 391125 = 45 13 Atomic Energy Education Society/Distance Learning Programme/2020

14. . Cube root of a cube number through Estimation method. If the given number is a cube number, then the estimation method can be used to find its cube root. Let us understand the steps of the estimation method . 14 Atomic Energy Education Society/Distance Learning Programme/2020

15. Find the cube root of the cube number 857375 Step 1 : Make groups of three digits starting from the right-most digit of the number. 857 375 second group first group Step 2 :First group, i.e., 375 will give the one s (or unit 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 its cube root ends in 5. So, we get 5 at the unit s place of the cube root. Step 3 : Now take another group, i.e., 857. We know that 93 = 729 and 103 = 1000. Also, 729 < 857 < 1000 . We take the one s place, of the smaller number 729 as the ten s place of the required cube root. So, 3857375 = 95 Atomic Energy Education Society/Distance Learning Programme/2020 15

16. So To find the cube root of a cube number by the estimation method : Make groups of three digits starting from the right-most digit of the given number. The First group gives the one s digit of the required cube root. The Second group gives the ten s digit of the required cube root. 16 Atomic Energy Education Society/Distance Learning Programme/2020

17. Another problem Find the cube root of 17576 Find the cube root of 17576 Step 1 : Form groups of three, starting from the rightmost digit of 17576 i.e., 17 576 In this case, one group i.e., 576 has three digits, whereas 17 has only two digits. Step 2 : Take 576. The digit 6 is at its one s place. We take the one s place of the required cube root as 6. Step 3 : Take the other group, i.e., 17. 17 lies between 8 and 27 i.e., 2 3 < 17 < 3 3 The smaller number among 2 and 3 is 2. Take 2 as ten s place of the cube root of 17576. Therefore 317576 = 26 17 Atomic Energy Education Society/Distance Learning Programme/2020

18. More problems .. Cube root of 50653 Cube root of 50653 1 Make groups of three digits from the right. 50 2nd group 1st group The first group will give us the unit s digit of the required cube root. The number 653 ends in 3. We know that 3 comes at the unit s place of a cube number only when its cube root ends in 7. So, we get 7 at the unit s place of the cube root. In the second group, 50 lies between 3 3 (=27) and 4 3 (=64) i.e., 3 3 < 50 < 4 3 Take the smaller number between 3 and 4, which is 3 . This will be the ten s place digit of the cube root. Therefore the cube root of 50653 is 37 ?????? = 37 653 18 Atomic Energy Education Society/Distance Learning Programme/2020

19. Practice Time Practice Time Find the Cube root of 110592 by prime factorisation method. Find the cube root of 46656 by estimation method. If the cube of 7 is 343, what is the cube root of 343? If the cube root of a number is 8, then what is the cube number? *^*^*^*^*^*^*^ 19 Atomic Energy Education Society/Distance Learning Programme/2020

20. What have we discussed? Meaning of cube root and its symbol. Finding cube root by prime factorisation method. Finding cube root of a cube number by estimation method. 20 Atomic Energy Education Society/Distance Learning Programme/2020

21. . End of Module 3 of 3 21 Atomic Energy Education Society/Distance Learning Programme/2020