Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Introduction to Cubes and Cube Roots ↓


Properties of Cube Numbers


Cube numbers are interesting and important in the world of mathematics, especially in class 8 where we start learning more about numbers, their roots and their various properties to build a strong foundation in mathematics. It is essential to understand cubes and cube roots as they appear frequently in algebra, geometry and other branches of mathematics.

What is a cube number?

A cube number is the result of multiplying an integer by itself twice, that is, raising it to the power of three. The word "cube" comes from the geometric shape of the cube, since the volume of a cube is found by cubing the length of one of its sides.

The formula for finding the cube of a number is as follows:

Cube of a number = n × n × n = n 3

For example:

  • The cube of 2 is: 2 × 2 × 2 = 8
  • Cube of 3 is: 3 × 3 × 3 = 27

Properties of cube numbers

1. When cubing positive integers the positive integer is always positive

Cubing a positive integer always gives a positive cube number. For example, the cube of 5, which is positive, will be positive. The calculation will be as follows:

5 × 5 × 5 = 125

2. Negative cube when cubing negative integers

When we cube a negative number, the result is a negative cube. This is because multiplying a negative number by itself gives a positive number, but when it is multiplied by a negative again (for the third time), the result becomes negative. Consider the cube of -2:

-2 × -2 × -2 = -8

3. Zero as a cube number

The cube of zero is also zero. This property is simple:

0 × 0 × 0 = 0

4. Observing patterns in cube numbers

When you start listing the cube numbers, certain patterns appear. These patterns often help in identifying cube numbers or calculating them mentally. Look at the sequence of cube numbers below:

First consider the cubes of some natural numbers:

  • 1 3 = 1
  • 2 3 = 8
  • 3 3 = 27
  • 4 3 = 64
  • 5 3 = 125

Note that as we move down the list of natural numbers, the difference between successive cube numbers increases.

5. Addition property of odd numbers

There is an interesting property connecting cubes to the sum of odd numbers. The cube of a natural number is the sum of a series of odd numbers. Consider this sequence for better understanding:

n 3 = 1 + 3 + 5 + ... + (2n-1)

For example, the cube of 4:

4 3 = 1 + 3 + 5 + 7 = 64

6. Cube root property

The cube root of a cubed number is the number whose root was cubed. This property is both straightforward and necessary when reversing a cubed operation.

For example, if 27 = 3 × 3 × 3, then the cube root of 27 is 3.

7. Visualization of cube numbers

Representing cube numbers geometrically involves looking at cubes of dimension n, where n represents the length of a side, and n 3 gives the volume of the cube.

N N N

The above illustration shows a cube with each side of length n, which indicates that the volume is calculated as n × n × n = n 3.

8. Perfect cube numbers

A number is a perfect cube if it is the cube of an integer. This distinguishes cube numbers that are exact from those that are not; for example, 27 is a perfect cube (3 3), while 26 is not. Understanding perfect cubes is helpful in solving equations and inequalities involving cube roots.

Examples of calculating cube numbers

Finding the cube of a number

Let's start by calculating the cube of simple integers using the basic formula:

Example 1: Find the cube of 6.

6 × 6 × 6 = 216

Example 2: What is the cube of 10?

10 × 10 × 10 = 1000

Decoding cube roots

On the other hand, we often need to find the cube root of a given number, which is virtually the same as undoing the cube operation.

Example 3: Find the cube root of 64.

∛64 = 4

Importance of cube numbers

Cubes and cube roots are associated with various aspects of mathematics and real-world applications. From determining volume in geometry to simplifying algebraic expressions and solving equations, they play a major role in the practical application of mathematical concepts.

Further exploration

Understanding cube numbers provides a great foundation for diving into more complex mathematical concepts. We encourage you to practice more with cube numbers, explore patterns, and apply them to solve more challenging problems as you advance in your study of mathematics.

Practising cube numbers and their properties is essential to master more advanced mathematical concepts that will be learned in higher classes.


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Properties of Cube Numbers

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