Cube and Cube Roots in One Shot | Maths - Class 8th | Umang | Physics Wallah

Introduction

This tutorial will guide you through the concepts of cubes and cube roots, which are essential parts of mathematics for Class 8 students. Understanding these concepts is crucial for solving various mathematical problems. This guide will outline the steps to calculate cubes, identify perfect cubes, and estimate cube roots effectively.

Step 1: Understanding the Cube of a Number

• The cube of a number is found by multiplying the number by itself two additional times.
• The formula to calculate the cube of a number ( n ) is: [ n^3 = n \times n \times n ]
• Example:
  • To find the cube of 3: [ 3^3 = 3 \times 3 \times 3 = 27 ]
• Practical Tip: Memorize cubes of numbers from 1 to 10 for quick reference, as they frequently appear in math problems.

Step 2: Identifying Perfect Cubes

• A perfect cube is a number that can be expressed as the cube of an integer.
• Examples of perfect cubes:
  • ( 1^3 = 1 )
  • ( 2^3 = 8 )
  • ( 3^3 = 27 )
  • ( 4^3 = 64 )
• Common Pitfall: Not all numbers are perfect cubes. For instance, 10 is not a perfect cube since there is no integer ( n ) such that ( n^3 = 10 ).

Step 3: Calculating Cube Roots

• The cube root of a number is the value that, when cubed, gives the original number.
• The symbol for cube root is ( \sqrt[3]{n} ).
• Example:
  • To find the cube root of 27: [ \sqrt[3]{27} = 3 \quad \text{(since ( 3^3 = 27 ))} ]
• Practical Tip: Use a calculator for larger numbers to find cube roots quickly.

Step 4: Estimating Cube Roots

• When the number is not a perfect cube, you can estimate the cube root.
• Steps to Estimate:
  • Identify the two nearest perfect cubes surrounding the number.
  • For example, to estimate ( \sqrt[3]{50} ):
    • 27 and 64 are the nearest perfect cubes (since ( 3^3 = 27 ) and ( 4^3 = 64 )).
    • Therefore, ( \sqrt[3]{50} ) is slightly more than 3 but less than 4.
• Practical Tip: You can use interpolation between the two perfect cubes for a more accurate estimate.

Conclusion

Understanding cubes and cube roots is fundamental in mathematics. Remember to:

• Calculate cubes using the formula ( n^3 ).
• Identify perfect cubes by recalling known cubes.
• Calculate cube roots and estimate them when necessary. By mastering these steps, you will enhance your mathematical skills and prepare yourself for more advanced concepts. Practice these techniques regularly to gain confidence in your calculations.