cubes and cube roots icse grade VIII

1. Ch 4 Cubes and Cube roots
 Perfect cubes
 Properties of cube numbers
 Prime factorisation to find cube root
 Imperfect cube to perfect cube
 Cube root of negative numbers
 Cube root of decimals
 Cube root of rational numbers

2. CUBE NUMBERS
AND ITS PROPERTIES

3. Property 1
If a number end with 1, then its cube will also end with 1
Example:
Numbers One’s Place Rule Answer
51 1 number end with 1 Cube of 51 will end with 1
121 1 number end with 1 Cube of 121 will end with 1
191 1 number end with 1 Cube of 191 will end with 1
23571 1 number end with 1 Cube of 23751 will end with 1

4. Property 2
If a number ends with 0 , 1 , 4 , 5 , 6 and 9 , then its cube will end with same number
For Example:
Numbers One’s Place Rule Answer
11 1 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 11 will end with 1
14 4 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 14 will end with 4
15 5 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 15 will end with 5
16 6 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 16 will end with 6
20 0 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 20 will end with 0
19 9 number ends with 0 , 1 , 4 , 5 , 6 and 9 Cube of 19 will end with 9

5. Property 3
If a number end with 2, then its cube will end with 8
If a number end with 8, then its cube will end with 2
For Example:
Numbers One’s place Rule Answer
12 2 Number ends with 2 Cube of 12 will end with 8
18 8 Number ends with 8 Cube of 8 will end with 2

6. Property 4
If a number end with 3, then its cube will end with 7
If a number end with 7, then its cube will end with 3
For Example:
Numbers Ones place Rule Answer
13 3 number end with 3 cube of 13 will end with 7
17 7 number end with 7 cube of 17 will end with 3

7. Property 5
Observe the following table
Cubes of an even
numbers are all even
numbers
Cubes of odd numbers
are all odd numbers
Number Square
2 8
4 64
6 216
8 512
10 1000
Number Square
1 1
3 27
5 125
7 343
9 729

8. Try These
By observing the units digits find which of the numbers cannot be a
perfect cube
2321 , 1335 , 20949 , 30550 , 21368 , 4173.

9. What are Perfect
Cubes?

10. A perfect cube is a number that is the cube of an
integer. For example, 125 is a perfect cube since
125 = 5 × 5 × 5 =
53. Some other examples of perfect cubes are 1, 8,
27, 64, 125, 216, 343, …
Perfect
Cube

11. What are Non-Perfect
Cubes?

12. There are many numbers which are not perfect
cubes and we cannot find the cube root of such
numbers using the prime factorisation and
estimation method. Let us find the cube root of 150
here. Clearly, 150 is not a perfect cube. It will be
around 5.31. So the cube root value of non perfect
cubes are in decimals or not integers.
Non-Perfect
Cube

13. What is Cube
Root?

14. In mathematics, a cube root of a number x is a
number y such that y³ = x. All nonzero real
numbers, have exactly one real cube root and a
pair of complex conjugate cube roots, and all
nonzero complex numbers have three distinct
complex cube roots.
Cube
Root

15. 3 27
CUBE
3 ✕ 3 ✕ 3
CUBE ROOT

17. What is Prime
Factorisation?

18. Prime
Factorisation
 Prime factorization of any given number is to breakdown the
number into its factors until all of its factors are prime numbers.
This can be achieved by dividing the given number from smallest
prime number and continue it until all its factors are prime.

19. Finding Cube Root by Prime
Factorisation

20. In order of finding cube root by prime factorization we use the
following
steps :
Step I : Obtain the given number.
Step II : Resolve it into prime factors.
Step III : Group the factors in 3 in such a way that each number
of the group is same.
Step IV : Take one factor from each group.
Step V : Find the product of the factors obtained in step IV. This
product is the required cube root.
Steps to Find Cube Root by Prime
Factorisation

21. Solved Example
#1

22. First of all we will factorise 1728 (as shown in the figure).
Now we will write all the prime factors obtained as - 1728 =
2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
Then we will make groups of three same numbers and write
one common of them as -
1728 = [2 × 2 × 2] × [2 × 2 × 2] × [3 × 3 × 3]
= 2 × 2 × 3
= 12
Therefore, cube root of 1728 is 12.
Find the cube root of
1728.

23. Solved Example #2

24. First of all we will factorise 970299 (as shown in the figure).
Now we will write all the prime factors obtained as -
970299 = 3 × 3 × 3 × 3 × 3 × 3 × 11 × 11 × 11
Then we will make groups of three same numbers and write
one
common of them as -
1728 = [3 × 3 × 3] × [3 × 3 × 3] × [11 × 11 × 11]
= 3 × 3 × 11
= 99
Therefore, cube root of 970299 is 99.
Find the cube root of 970299.