Squares & Square Roots
Perfect Squares & Their Properties
A perfect square is a number that is the square of a whole number, like 1, 4, 9, 16, 25. A perfect square can only end in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8.
Example 1: What is 12²?
12 × 12 = 144.
Example 2: Can a number ending in 7 be a perfect square?
No — perfect squares never end in 2, 3, 7 or 8.
Quick recap
- Perfect square = the square of a whole number.
- Perfect squares end only in 0, 1, 4, 5, 6 or 9.
✓ Quick check
Which of these is a perfect square?
64 = 8², a perfect square.
A perfect square can never end in which digit?
Perfect squares never end in 2, 3, 7 or 8 — so 8.
Finding Squares & Square Roots
The square root of a number is the value that, when multiplied by itself, gives that number. We can find it by prime factorisation (pairing factors) or by long division. For example, √196 = 14 and √225 = 15.
Example 1: Find √196.
14, because 14 × 14 = 196.
Example 2: Find √81.
9, because 9 × 9 = 81.
Quick recap
- Square root: the number that multiplies by itself to give the value.
- Use prime factorisation (pairing) or long division.
✓ Quick check
What is √169?
13 × 13 = 169.
What is √256?
16 × 16 = 256.
Pythagorean Triplets
A Pythagorean triplet is three whole numbers a, b, c with a² + b² = c². Examples are (3, 4, 5), (5, 12, 13) and (8, 15, 17). For any whole number m, (2m, m² − 1, m² + 1) is a triplet.
Example 1: Is (6, 8, 10) a Pythagorean triplet?
6² + 8² = 36 + 64 = 100 = 10², so yes.
Example 2: Find √625.
25, because 25 × 25 = 625.
Quick recap
- A triplet satisfies a² + b² = c².
- Common triplets: (3,4,5), (5,12,13), (8,15,17).
✓ Quick check
Which is a Pythagorean triplet?
5² + 12² = 25 + 144 = 169 = 13².
What is √400?
20 × 20 = 400.
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