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


In math, every number has different relationships with other numbers. Some of these relationships may be simple, while others may require a little more thought to understand. Today, we will explore "cube" and "cube root," which are essential concepts in understanding these numerical relationships.

What is a cube?

The cube of a number is a number multiplied by itself twice. This essentially raises the number to the power of three. Imagine you have a number, call it n. The cube of n, represented as n 3, is calculated as follows:

 n × n × n = n 3

This operation is called "cubing" the number.

Example of a cube

Let's find the cube of 4:

 4 × 4 × 4 = 64

Therefore the cube of 4 is 64.

Visualization of cubes

To help us visualize cubes, imagine a three-dimensional box or cube. A geometric cube has all sides of the same length. If each side of the cube is of length n, then the volume of the cube will be n 3.

Each side of this cube has length n. The volume of the cube is calculated as the space present inside the cube, which is expressed as n × n × n.

Normal cubes

For practice it's helpful to remember some basic cubes:

  • 1 3 = 1
  • 2 3 = 8
  • 3 3 = 27
  • 4 3 = 64
  • 5 3 = 125
  • 6 3 = 216
  • 7 3 = 343
  • 8 3 = 512
  • 9 3 = 729
  • 10 3 = 1000

What is cube root?

The cube root is the inverse operation of cubing a number. It is the number that when multiplied by itself three times gives the original number. The cube root of a number n is written as ∛n.

Cube root example

What is the cube root of 27?

We get that 3 multiplied by itself twice gives 27:

 3 × 3 × 3 = 27

Therefore, ∛27 = 3.

Calculating the cube root

As shown in the previous example, calculating the cube root for perfect cubes can be straightforward. Let's practice calculating the cube root of some common perfect cubes:

  • ∛1 = 1
  • ∛8 = 2
  • ∛27 = 3
  • ∛64 = 4
  • ∛125 = 5
  • ∛216 = 6
  • ∛343 = 7
  • ∛512 = 8
  • ∛729 = 9
  • ∛1000 = 10

Finding cube roots of imperfect cubes

Not all numbers are perfect cubes, which can make finding their cube root a little tricky. Approximation or a calculator can help find the cube root of such numbers.

Example of incomplete cube root

Let's find the cube root of 20 approximately:

Since 2 3 = 8 and 3 3 = 27, we know that ∛20 lies between 2 and 3 Using successive approximations, ∛20 ≈ 2.71.

Practical applications of cube and cube root

Cubes and cube roots are not just theoretical concepts, they have a lot of practical importance, especially in fields such as engineering, architecture and even nature. For example, understanding the concept of volume is very important for designing buildings, packaging and various scientific calculations.

Real-world example

Imagine that you are tasked with determining the size of a water tank that can hold exactly 1,000 cubic meters of water. Knowing that the container must be a cube, we can calculate the required side length:

 ∛1000 = 10

The length of the side of the cube should be 10 meters.

Reviews and exercises

Practice is very important to understand cubes and cube roots properly. Here are some exercises to strengthen your understanding:

  1. Find the cube of the number 11.
  2. Find the cube root of 729.
  3. Find the cube root of 50 approximately.
  4. Find the volume of a cube whose side length is 7 units.
  5. If n 3 = 512 then find the solution for n.

By understanding the principles of cubes and cube roots, you lay the foundation for more advanced mathematical concepts. Whether calculating volumes, solving equations, or understanding real-world problems, these operations play a vital role in mathematics and its applications.


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Introduction to Cubes and Cube Roots

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