Class 8 → Mathematics → Chapter 1: A Square and a Cube (New Course)

I. Chapter Summary

This chapter introduces students to the concepts of squares and cubes of numbers, along with their properties and patterns. It focuses on understanding how numbers behave when multiplied by themselves (square) or thrice (cube). Students learn methods to identify perfect squares and cubes, find square roots and cube roots, and observe interesting number patterns. These concepts form the foundation for algebra, mensuration, and higher mathematical problem-solving.

II. Key Concepts Covered

1. Square of a Number

• The square of a number is obtained by multiplying the number by itself.
• Example: $5^2 = 5 \times 5 = 25$

2. Perfect Squares

• Numbers that are squares of integers.
• Example: 1, 4, 9, 16, 25, etc.

3. Properties of Perfect Squares

• A perfect square has an even number of prime factors.
• It always ends with digits: 0, 1, 4, 5, 6, or 9.

4. Square Roots

• The inverse operation of squaring.
• Example: $\sqrt{25} = 5$

5. Cube of a Number

• The cube of a number is obtained by multiplying the number three times.
• Example: $3^3 = 27$

6. Perfect Cubes

• Numbers that are cubes of integers.
• Example: 1, 8, 27, 64, 125, etc.

7. Cube Roots

• The inverse operation of cubing.
• Example: $\sqrt[3]{27} = 3$

8. Patterns in Squares and Cubes

• Squares and cubes follow unique patterns helpful in quick calculations.

III. Important Questions

(A) MCQs (1 Mark)

1. Which of the following is a perfect square?
   (a) 20 (b) 25 (c) 30 (d) 35
   Answer: (b) 25
2. The square root of 144 is:
   (a) 10 (b) 11 (c) 12 (d) 13
   Answer: (c) 12
3. Which of the following is a perfect cube?
   (a) 16 (b) 27 (c) 36 (d) 45
   Answer: (b) 27
4. The cube root of 125 is:
   (a) 4 (b) 5 (c) 6 (d) 7
   Answer: (b) 5

(B) Short Answer Questions (2/3 Marks)

1. Find the square of 18.
2. Check whether 256 is a perfect square.
3. Find the cube of 7.
4. Determine the cube root of 343.

(C) Long Answer Questions (5 Marks)

1. Explain the method to find the square root using prime factorization.
2. Verify whether 729 is a perfect cube using prime factorization.
3. Find the square root of 2025 using long division method.
4. Explain patterns observed in squares of natural numbers.

(D) HOTS Questions

1. A number when squared gives 784. What will be the result when the number is cubed?
2. Find the smallest number that must be multiplied with 108 to make it a perfect cube.

IV. Key Formulas / Concepts

• Square: $n^2 = n \times n$
• Cube: $n^3 = n \times n \times n$
• Square Root: $\sqrt{n}$​
• Cube Root: $\sqrt[3]{n}$​

Example:

• $6^2 = 36, \quad 6^3 = 216$

V. Deleted Portions (CBSE 2025–2026)

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Note: This is an estimate. Actual marks distribution may vary.

VII. Previous Year Questions (PYQs)

1 Mark

• Find the square of 13. (CBSE 2020)

2/3 Marks

• Find the square root of 169 using factorization. (CBSE 2019)

5 Marks

• Explain how to find cube roots using prime factorization with an example. (CBSE 2018)

VIII. Real-World Application Examples

• Area Calculation: Square helps in finding area of square-shaped objects like tiles.
• Volume Calculation: Cubes are used in finding volume of objects like boxes.
• Engineering & Construction: Used in measurements and design.
• Computer Science: Powers and exponents are widely used in algorithms.

IX. Student Tips & Strategies for Success

Time Management

• Practice 10–15 problems daily.
• Allocate time for revision weekly.

Exam Preparation

• Learn squares up to 30 and cubes up to 20.
• Practice different methods (factorization, division).

Stress Management

• Take short breaks.
• Practice regularly to build confidence.

X. Career Guidance & Exploration (Class 8 Level)

• Builds foundation for Mathematics, Engineering, Data Science.
• Helps in logical thinking and problem-solving.
• Important for future streams:
  • Science → Engineering, Physics
  • Commerce → Accounts, Statistics
  • Arts → Economics, Data Analysis

XI. Important Notes

• Always refer to the latest NCERT textbooks.
• Practice regularly for better understanding.
• Focus on concepts rather than rote learning.
• Solve previous year questions for exam readiness.