Squares and Square Roots
Focused last-minute revision with key concepts, formulas, worked examples, exam traps and practice.
Chapter Snapshot
- A perfect square is n² for an integer n.
- Perfect squares cannot end in 2, 3, 7 or 8.
- Prime factors in a perfect square occur with even exponents.
- Square root reverses squaring.
- Pythagorean triplets satisfy a²+b²=c².
What You Will Learn
- Recognise perfect squares
- Use quick properties of square numbers
- Find square roots by prime factorisation
- Understand the long-division method
- Generate simple Pythagorean triplets
Last-Minute Chapter Map
Use this chapter in three passes:
- Understand the core ideas.
- Memorise formulas and exam traps.
- Attempt the compact practice set without looking at the answers.
Perfect squares
A perfect square is the product of an integer with itself. Common squares to memorise include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and 225.
Quick checks
Squares of even numbers are even and squares of odd numbers are odd. A perfect square cannot end in 2, 3, 7 or 8. Trailing zeros in a perfect square occur in an even count.
Prime factorisation
Write the number as a product of primes and pair equal factors. Take one factor from each pair to obtain the square root.
Long division
Pair digits from the decimal point outward, choose the largest fitting square, subtract, bring down the next pair, double the current quotient and determine the next digit.
Pythagorean triplets
Three positive integers form a Pythagorean triplet when a²+b²=c². A useful generator is 2m, m²-1, m²+1 for integer m greater than 1.
Formula Sheet
Formula / Key Relations
Square
Definition of a square.
Pythagorean relation
Used for right-triangle number triplets.
Triplet generator
m is an integer greater than 1.
Worked Examples
Pair equal prime factors.
The triplet is 8, 15, 17.
Exam Traps
Quick Revision
Quick Revision
- A perfect square is n² for an integer n.
- Perfect squares cannot end in 2, 3, 7 or 8.
- Prime factors in a perfect square occur with even exponents.
- Square root reverses squaring.
- Pythagorean triplets satisfy a²+b²=c².
- Revise the formula sheet once more before attempting questions.
Compact Practice
Which cannot be a perfect square: 64, 81, 98, 121?
View Solution
98
√225?
View Solution
15
Is 2025 a perfect square?
View Solution
Yes, 45².
Generate a Pythagorean triplet for m=5.
View Solution
10, 24, 26
Final Chapter Summary
Chapter Summary
- A perfect square is n² for an integer n.
- Perfect squares cannot end in 2, 3, 7 or 8.
- Prime factors in a perfect square occur with even exponents.
- Square root reverses squaring.
- Pythagorean triplets satisfy a²+b²=c².
- Prioritise correct method, clean working and final units where applicable.