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Squares and Square Roots

Gujarat Board · Class 8 · Mathematics

Quick revision notes for Squares and Square Roots — Gujarat Board Class 8 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

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Key Topics to Revise

1

Square Numbers and Perfect Squares

  • A square number (or perfect square) is a number that can be expressed as n² where n is a natural number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
  • The first 20 perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
  • If a number cannot be expressed as n² for any natural number n, it is NOT a perfect square.
2

Properties of Square Numbers

  • UNITS DIGIT RULE: A perfect square can only end in 0, 1, 4, 5, 6, or 9. It can NEVER end in 2, 3, 7, or 8.
  • UNITS DIGIT PATTERN: If a number ends in 1 or 9, its square ends in 1. If it ends in 2 or 8, its square ends in 4. If it ends in 3 or 7, its square ends in 9. If it ends in 4 or 6, its square ends in
  • TRAILING ZEROS: A perfect square always has an even number of zeros at its end. For example, 100 (2 zeros), 10000 (4 zeros). A number with an odd number of zeros (like 1000) is NOT a perfect square.
3

Finding the Square of a Number

  • For any two-digit number, split it as (a + b) where a is a multiple of 10 and b is the units digit, then use (a+b)² = a² + 2ab + b².
  • Special shortcut for numbers ending in 5: If a number ends in 5, its square = (preceding digits × next number) followed by 25. Example: 65² = (6×7) hundreds + 25 = 4225.
  • Product of two consecutive odd/even numbers: (a-1)(a+1) = a² - 1. Example: 19 × 21 = 20² - 1 = 399.
4

Pythagorean Triplets

  • A Pythagorean triplet is a set of three natural numbers (a, b, c) such that a² + b² = c², where c is the largest.
  • Common Pythagorean triplets to memorise: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (6,8,10), (9,40,41).
  • General formula: For any natural number m > 1, the triplet is (2m, m²-1, m²+1).

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Full Notes

Key Concepts

A square number is a naturalSquare numbers have distinct characteristicsSeveral interesting patterns exist in squareUsing algebraic identitiesA Pythagorean triplet consists of three

Frequently Asked Questions

What are the important topics in Squares and Square Roots for Gujarat Board Class 8 Mathematics?
Key topics in Squares and Square Roots include Squares and Square Roots – Chapter Overview, Squares and Square Roots - Concept Overview, Squares and Square Roots – Complete Chapter Overview. These are the concepts Gujarat Board Class 8 examiners draw on most — study them first, then practise related questions.
How to score full marks in Squares and Square Roots — Gujarat Board Class 8 Mathematics?
Understand the core concepts first, then work through the 45 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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