Squares and Square Roots Notes
A perfect square is the product of an integer with itself. This chapter covers properties of square numbers, patterns, finding squares quickly, Pythagorean triplets, and finding square roots by prime factorisation and by long division.
- 3 min read
- 10 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1.Square Numbers and Their Properties
- 2.Patterns and Quick Squaring
- 3.Pythagorean Triplets
- 4.Finding Square Roots
- 1 key formulas
- 1 concept checks
Square Numbers and Their Properties
The square of a number n is n × n, written n². The numbers 1, 4, 9, 16, 25, … are perfect squares.
Useful properties: a perfect square never ends in 2, 3, 7, or 8. A perfect square ends in an even number of zeros. Squares of odd numbers are odd; squares of even numbers are even. The square of a number ending in 5 ends in 25.
The number of non-perfect-square numbers between n² and (n + 1)² is 2n. For example, between 9 (= 3²) and 16 (= 4²) there are 2 × 3 = 6 numbers.
Patterns and Quick Squaring
The sum of the first n odd natural numbers is n². So 1 + 3 + 5 + 7 = 16 = 4².
For a number ending in 5, say a5: its square is a(a + 1) followed by 25. For example 35² = (3 × 4) | 25 = 1225, and 85² = (8 × 9) | 25 = 7225.
The identity (a + b)² = a² + 2ab + b² lets you square numbers close to a round number: 103² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609.
Pythagorean Triplets
Three natural numbers a, b, c with a² + b² = c² form a Pythagorean triplet — the sides of a right-angled triangle. Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).
For any natural number m > 1, the triplet (2m, m² − 1, m² + 1) is Pythagorean. Taking m = 4 gives (8, 15, 17).
Multiplying a triplet by a constant gives another triplet, e.g. (6, 8, 10) from (3, 4, 5).
Finding Square Roots
By prime factorisation: write the number as a product of primes, pair equal primes, and take one prime from each pair. For 324 = 2 × 2 × 3 × 3 × 3 × 3, √324 = 2 × 3 × 3 = 18.
By long division: this method also works for numbers that are not perfect squares and for decimals, giving the square root to any required number of places.
To make a number a perfect square, divide or multiply by the prime that is unpaired. For example, 90 = 2 × 3 × 3 × 5; multiplying by 2 × 5 = 10 gives 900 = 30².
Squares and Square Roots: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Square Numbers and Their Properties
- Patterns and Quick Squaring
- Pythagorean Triplets
- Finding Square Roots
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (1)
Pythagorean Triplets
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
How can I tell quickly that a number is not a perfect square?
If it ends in 2, 3, 7, or 8, or ends in an odd number of zeros, it cannot be a perfect square.
What is the difference between the prime factorisation method and the long division method for square roots?
Prime factorisation is quick for perfect squares. Long division works for any number, including non-perfect squares and decimals, and gives the root to as many decimal places as needed.
How do I generate a Pythagorean triplet?
Pick a natural number m > 1. Then (2m, m² − 1, m² + 1) is a Pythagorean triplet, because (2m)² + (m² − 1)² = (m² + 1)².