In this page we will explain the topics for the chapter 6 of Cube and Cube Roots Class 8 Maths.We have given quality notes and video to explain various things so that students can benefits from it and learn maths in a fun and easy manner, Hope you like them and do not forget to like , social share
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Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be
expressed as sum of two cubes in two different ways.
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
1729 is the smallest Hardy– Ramanujan Number. There are an infinitely many such numbers. Few are 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), Check it with the numbers given in the brackets
Cube Number
Numbers obtained when a number is multiplied by itself three times are known as cube numbers
Example
1=13
8=23
27=33
Or if a natural number m can be expressed as n3 where n is also a natural number, then m is a cube number
The numbers 1, 8, 27, 125 ... are cube numbers. These numbers are also called perfect cubes.
Some Important point to Note
Prime Factorization of Cubes
When we perform the prime factorization of cubes number, we find one special property
8= 2×2×2 (Triplet of prime factor 2)
216 = (2 × 2 × 2) × (3 × 3 × 3) ( Triplet of 2 and 3)
Each prime factor of a number appears three times in the prime factorization of its cube.
Smallest multiple that is a perfect cube
Here we find the prime factorization of the number. Then we find the prime factor required to make all of them in triplet.
Example
Find the smallest multiple that will make 392 perfect cube Solution:
392 = 2 × 2 × 2 × 7 × 7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect ube. To make its a cube, we need one more 7. In that case
392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube
Cube Root
Cube root of a number is the number whose cube is given number
So we know that
27=33
Cube root of 27
$\sqrt[3]{27} =3 $
How to Find Cube root
Finding cube root through prime factorization
This method, we find the prime factorization of the number.
We will get same prime number occurring in pair for perfect square number. Square root will be given by multiplication of prime factor occurring in pair
Consider
1331
1331= (11×11×11)
$\sqrt[3]{1331} =11 $
This can be well explained with the example
The given number is 17576. Step 1 Form groups of three starting from the rightmost digit of 17576.
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.
Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.
The smaller number among 2 and 3 is 2.
The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of
17576.
Thus,
$\sqrt[3]{17576} =26$
Extra Zing
1) for any Positive integer m, $m^3 > m^2$ i.e cube is greater than square
2) For any negative integer m, $m^3 < m^2$ i.e cube is less than square , as the cube is always negative number and square is positive number
3) Cubes can be written as Addition consecutive odd numbers
Frequently asked Questions on CBSE Class 8 Maths Chapter 7: Cubes and Cube Roots
What is cube root definition Class 8?
Cube root of a number is the number when multiplied itself by thrice gives the original value
Example Cube root of 8 is 2 as when 2 is multipled by itself thrice, it gives the Original value
$2 \times 2 \times 2=8$
What is a cube root of 729?
Cube root of a number can be found using prime factorization method
$729= 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$\sqrt[3]{729}=3 \times 3 =9$
Is 10000 is a perfect cube?
Cube root of a number can be found using prime factorization method
$10000= 10 \times 10 \times 10 \times 10$
It is clear 10000 is not a perfect cube
Is 512 is a perfect cube?
Cube root of a number can be found using prime factorization method
$512= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $
$\sqrt[3]{729}=2 \times 2 \times 2=8$
What is the formula of cube and cube root?
Cube of a number is given by $n \times \times n =n^3$
Example Cube root of 2 is $2 \times 2 \times 2=8$
Cube root in inverse of Cube and its formula is $\sqrt[3]{n}$
What is the properties of cube?
Properties of Cube of Number:
(i) Cubes of even number are even whereas Cubes of odd numbers are odd.
(ii) The sum of the cubes of first n natural numbers is equal to the square of their sum.
(iii) The below properties will help in finding the Unit digits of the cubes
