Squares and Square Roots
1. Perfect Squares
A perfect square is the square of an integer.
0² = 0, 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225, 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400, 25² = 625, 30² = 900, 40² = 1600, 50² = 2500.
'Memorise squares up to 25 — they appear in EVERY chapter of mathematics.'
2. Properties of Square Numbers
| Property | Example |
|---|---|
| A number ending in 2, 3, 7, or 8 is NEVER a perfect square | 22, 33, 47, 58 are NOT perfect squares |
| A number ending in an ODD number of zeros is NEVER a perfect square | 1000, 250, 40 are NOT perfect squares |
| Square of an EVEN number is EVEN | 12² = 144 (even) |
| Square of an ODD number is ODD | 13² = 169 (odd) |
| Square of a number ending in 5 ends in 25 | 35² = 1225, 45² = 2025 |
| Square of a number ending in 1 or 9 ends in 1 | 11² = 121, 19² = 361 |
| Square of a number ending in 4 or 6 ends in 6 | 14² = 196, 16² = 256 |
Pattern: (n + 1)² = n² + n + (n + 1)
Example: If 15² = 225, then 16² = 225 + 15 + 16 = 256. ✓
3. Square Roots by Prime Factorisation
To find the square root of a perfect square:
- Write the number as a product of its prime factors.
- GROUP the factors into PAIRS of identical factors.
- Take ONE factor from each pair.
- MULTIPLY those factors.
Worked Example: Find √144
144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3² Pair: (2×2), (2×2), (3×3) √144 = 2 × 2 × 3 = 12
Worked Example: Find √1764
1764 = 2 × 2 × 3 × 3 × 7 × 7 = 2² × 3² × 7² √1764 = 2 × 3 × 7 = 42
4. Square Roots by Long Division Method
Used for LARGER numbers where prime factorisation is difficult.
Worked Example: Find √4489 by long division.
Step 1: Place bars on the digits in PAIRS from the RIGHT. 44 | 89
Step 2: Find the largest number whose square ≤ 44. 6² = 36. Write 6 as quotient and divisor.
Step 3: Subtract 36 from 44. Remainder = 8. Bring down 89. New dividend = 889.
Step 4: DOUBLE the quotient (6 × 2 = 12). Find digit d such that 12d × d ≤ 889. d = 7 because 127 × 7 = 889.
Step 5: Subtract. Remainder = 0.
Therefore √4489 = 67.
Worked Example: Find √5579.56 (decimal).
Follow the same steps. Place bars on INTEGER part from RIGHT and DECIMAL part from LEFT.
5. Pythagorean Triplets
Three natural numbers (a, b, c) such that a² + b² = c².
Generating Triplets
For any natural number m > 1: (2m, m² — 1, m² + 1)
| m | Triplet | Check |
|---|---|---|
| 2 | (4, 3, 5) | 4² + 3² = 16 + 9 = 25 = 5² |
| 3 | (6, 8, 10) | 6² + 8² = 36 + 64 = 100 = 10² |
| 4 | (8, 15, 17) | 8² + 15² = 64 + 225 = 289 = 17² |
| 5 | (10, 24, 26) | 10² + 24² = 100 + 576 = 676 = 26² |
| 6 | (12, 35, 37) | 12² + 35² = 144 + 1225 = 1369 = 37² |
'Pythagorean triplets are useful in geometry problems involving right triangles.'
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| '√(a + b) = √a + √b' | FALSE! √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7 |
| '625 is not a perfect square' | 625 = 25²! Check the last two digits: 25 → could be a perfect square |
| 'A number ending in 4 cannot be a perfect square' | It CAN: 4, 64, 144, 324 are all perfect squares ending in 4 |
| 'Leaving out a factor when pairing' | Ensure EVERY prime factor is paired. Unpaired factors mean the number is NOT a perfect square |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Identifying perfect squares, finding square roots of small numbers
- 3-mark questions: Prime factorisation method for square roots
- 4-mark questions: Long division method or Pythagorean triplet problems
- 6-mark questions: Application — area problems using square roots
Self-Test
Q1. Is 1568 a perfect square? Justify. A1. 1568 = 2⁵ × 7². The factors are 2² × 2² × 7² × 2. One '2' is unpaired. So 1568 is NOT a perfect square.
Q2. Find √3136 by prime factorisation. A2. 3136 = 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 = 2⁶ × 7². √3136 = 2 × 2 × 2 × 7 = 56.
Q3. Find the smallest number by which 720 must be MULTIPLIED to get a perfect square. A3. 720 = 2⁴ × 3² × 5. '5' is unpaired. Multiply by 5: 720 × 5 = 3600 = 60².
Q4. Find √53,824 by long division. A4. 5,38,24. Step-by-step: √53824 = 232. (Check: 232² = 53824)
Q5. Write a Pythagorean triplet whose smallest member is 8. A5. If 2m = 8, then m = 4. Triplet: (2×4, 4²—1, 4²+1) = (8, 15, 17). Check: 8² + 15² = 64 + 225 = 289 = 17² ✓.
Q6. The area of a square field is 8281 m². Find its side. A6. Side = √8281 = 91 m (by long division: 91² = 8281).
