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Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board) PDF Download

Introduction 


Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)This story is about S. Ramanujan, one of India’s greatest mathematical minds. When Prof. G.H. Hardy once visited him in a taxi with the number 1729, Hardy called the number "dull." 

Ramanujan quickly corrected him, noting that 1729 is actually the smallest number expressible as the sum of two cubes in two different ways. 

  • The cube of 10 plus the cube of 9: (103 + 93 = 1729)
  • The cube of 12 plus the cube of 1: (123 + 1= 1729)
    Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

This number, now known as the Hardy–Ramanujan Number, had been recognized long before, but Ramanujan's deep love for numbers led him to discover such fascinating properties.

Cubes

Word cube is used in geometry. It is a solid figure which all sides are equal.
Consider a cube of side 3 cm , How many cubes can be made of side 1cm from 3cm side cube!

Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

Cube of any number is obtained when the number is multiplied 3 times in a row.Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)Cubes till 10

Perfect Cube:  A perfect cube is a number that you get when you multiply a whole number by itself two more times.

A number of the form n3 where is an integer.  


Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

8, 27, 64 are perfect cubes.

There are only ten perfect cubes from 1 to 1000. Cube of any odd integer is odd and cube of any even integer is even.

Some Interesting Patterns

1.  Adding Consecutive Odd Numbers

Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

Number of consecutive odd number to add to get "n3" = n

2. Cubes and Their Prime Factors
Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

216 = 2 x 2 x 2 x 3 x 3 x3
Since each factor appears 3 times. 216 = (23 × 33) = 6 because an x bn = (a × b)n 

Example: Is 354 a perfect cube?

Sol: 354= 2×3×59
In the above factorization 2×3×59 remain.

Therefore, 354 is not a perfect cube.

Smallest Multiple that is a perfect cube

Example : Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube.

 Sol: 392 = 2 × 2 × 2 × 7 × 7 The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect cube. 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. Hence the smallest natural number by which 392 should be multiplied to make a perfect cube is 7.

Cube Roots

Lets understand an example -
There is a cube of volume 125 cm3 . Since all the side of cube is equal we want to find the length of our cube.
Volume of cube = (side of cube)3 unit3. ------------------------------------(1)
Side of cube = (Volume of cube)1/3.

Hence the side of any cube is nothing but a cube root of the volume of that cube.
How to denote cube root in mathematics.
Let's understand:  (25)1/3 = (53)1/3 = 5. 

Hence let's take an integer x then Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

Cube Root Through Prime Factorization Method

Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)

Example :  Find the cube root of 8000.

 Sol: Prime factorization of 8000 is 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 

So, Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)  = 2 × 2 × 5 = 20

What we have discussed

  • Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways. 
  •  Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, ... etc. 
  • If in the prime factorization of any number each factor appears three times, then the number is a perfect cube. 
  • The symbol  Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)  denotes cube root. For example Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board)
The document Indices and Cube root Chapter Notes | Mathematics Class 8 (Maharashtra Board) is a part of the Class 8 Course Mathematics Class 8 (Maharashtra Board).
All you need of Class 8 at this link: Class 8
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FAQs on Indices and Cube root Chapter Notes - Mathematics Class 8 (Maharashtra Board)

1. What is a cube and how is it defined in mathematics?
Ans.A cube is a three-dimensional geometric figure with six equal square faces. In mathematics, a cube of a number is the result of multiplying that number by itself twice. For example, the cube of 2 is calculated as 2 × 2 × 2, which equals 8. This concept is fundamental in geometry and algebra.
2. How do you find the cube root of a number?
Ans.The cube root of a number is the value that, when multiplied by itself twice, gives the original number. It is denoted by the radical symbol ∛ or by raising the number to the power of one-third. For example, the cube root of 27 is 3, since 3 × 3 × 3 = 27.
3. What is the prime factorization method for finding cube roots?
Ans.The prime factorization method involves breaking down a number into its prime factors. To find the cube root using this method, you express the number as a product of its prime factors, group the factors in sets of three, and then multiply one factor from each group. For example, to find the cube root of 216, the prime factorization is 2³ × 3³, and the cube root is 2 × 3 = 6.
4. Why is understanding cube roots important in mathematics?
Ans.Understanding cube roots is important because they are used in various mathematical applications, including solving equations, geometry, and real-world problems involving volume and three-dimensional shapes. It also helps in understanding higher-level concepts in algebra and helps in simplifying expressions.
5. Can you provide examples of cube numbers and their corresponding cube roots?
Ans.Cube numbers are integers that are cubes of other integers. Examples include 1³ = 1, 2³ = 8, 3³ = 27, and 4³ = 64. The corresponding cube roots are 1, 2, 3, and 4 respectively. These examples illustrate the relationship between numbers and their cubes, which is essential for mastering the concepts of cubes and cube roots.
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