Squares and Square Roots — Class 8 Mathematics
1. Perfect Squares
A PERFECT SQUARE is a number that is the SQUARE of an integer. n² = n × n. Examples: 1²=1, 2²=4, 3²=9, ..., 10²=100. First 20 perfect squares memorize them: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
Properties of Perfect Squares
- Last digit: A perfect square can end ONLY in 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8 → NOT a perfect square.
- Number of zeros: A perfect square ending in zeros has an EVEN number of trailing zeros. 100 (2 zeros) ✓. 1000 (3 zeros) ✗.
- Sum of first n odd numbers = n²: 1 = 1². 1+3 = 4 = 2². 1+3+5 = 9 = 3². 1+3+5+7 = 16 = 4². 'The sum of the first n ODD natural numbers is always a perfect square — n².'
- Between n² and (n+1)²: There are 2n non-perfect-square numbers. Example: Between 3² and 4² (between 9 and 16), there are 2×3 = 6 numbers: 10, 11, 12, 13, 14, 15. None is a perfect square.
- Product property: If a and b are perfect squares, a×b is also a perfect square.
2. Finding Square Roots
The SQUARE ROOT of a number n is a number m such that m² = n. Symbol: √n. √n is ALWAYS non-negative (principal square root).
Method 1 — Prime Factorisation
Express the number as a product of PRIME FACTORS. Make PAIRS of identical factors. Take ONE factor from each pair. MULTIPLY.
Example — √1764: 1764 = 2×882 = 2×2×441 = 2²×3×147 = 2²×3×3×49 = 2²×3²×7². Take one from each pair: 2×3×7 = 42. √1764 = 42. Check: 42² = 1764 ✓.
Example — √2025: 2025 = 3×675 = 3×3×225 = 3²×3×75 = 3²×3×3×25 = 3⁴×5². Take: 3²×5 = 9×5 = 45. √2025 = 45.
Method 2 — Long Division Method
Used for LARGE numbers and numbers that are NOT perfect squares. Steps:
- Place a BAR over every PAIR of digits starting from the right (units). For decimals, pair from decimal point outward.
- Find the LARGEST number whose square ≤ the first group. Write it as quotient AND divisor.
- SUBTRACT and BRING DOWN the next pair.
- DOUBLE the current quotient — this is the trial divisor.
- Find the next digit d such that (trial divisor + d) × d ≤ current remainder.
- Repeat until remainder = 0 (perfect square) or desired decimal places.
Example — √529 by Long Division: Pair: 5 29. Largest square ≤ 5: 2²=4. Remainder = 1. Bring down 29: 129. Trial divisor = 2×2 = 4. Find d: (40+d)×d ≤ 129. d=3: 43×3 = 129. Perfect! √529 = 23.
Example — √1369: Pair: 13 69. Largest square ≤ 13: 3²=9. Remainder = 4. Bring down 69: 469. Trial = 2×3 = 6. Find d: (60+d)×d ≤ 469. d=7: 67×7 = 469. √1369 = 37.
3. Pythagorean Triplets
Three natural numbers a, b, c such that a² + b² = c². For any natural number m > 1: a = 2m, b = m²−1, c = m²+1. This generates a Pythagorean triplet.
Example — m=3: a = 6, b = 9−1 = 8, c = 9+1 = 10. Check: 6²+8² = 36+64 = 100 = 10² ✓. (6, 8, 10) is a Pythagorean triplet. Example — m=4: a = 8, b = 16−1 = 15, c = 16+1 = 17. (8, 15, 17).
'Common triplets to memorise: (3,4,5), (5,12,13), (6,8,10), (7,24,25), (8,15,17), (9,40,41). These appear frequently in geometry problems.'
4. Square Roots of Decimals
Method 1 — Convert to fraction: √1.44 = √(144/100) = √144/√100 = 12/10 = 1.2. Method 2 — Long Division: Pair digits from decimal point outward in both directions. 7.29 → 7 . 29. √7.29 = 2.7.
5. Estimating Square Roots
When a number is NOT a perfect square, its square root is IRRATIONAL. Estimate using the nearest perfect squares. Example: √50. Nearest perfect squares: 7²=49, 8²=64. So √50 is between 7 and 8, closer to 7. Refine: 7.07²≈49.98, 7.08²≈50.13. √50 ≈ 7.07.
6. Common Mistakes
- √(a+b) ≠ √a + √b: √(9+16) = √25 = 5. √9+√16 = 3+4 = 7 ≠ 5. 'Square root of a SUM is NOT the sum of square roots.'
- √(a×b) = √a × √b: This IS valid. √(36×25) = √900 = 30. √36×√25 = 6×5 = 30.
- Last digit shortcut misuse: A number ending in 1 may OR may not be a perfect square. 11 is not a square. 121 is.
- Forgetting the ± sign in equations: x² = 25 → x = ±5. But √25 = +5 (always the principal root).
7. AP Exam Focus
| Topic | Marks |
|---|---|
| Prime factorisation method | 3-4 |
| Long division method | 4-5 |
| Pythagorean triplets | 2-3 |
| Square root of decimals | 3-4 |
Key Exam Tips
- Long division is the PREFERRED method for finding square roots of large numbers in exams.
- For 'find the smallest number to multiply by to get a perfect square': do prime factorisation, add the missing factors to complete pairs.
- Pythagorean triplets: use the formula (2m, m²−1, m²+1). State clearly which value of m you used.
Quick Self-Test
- Is 1521 a perfect square? (Answer: Check ending: 1 ✓. √1521 = 39. Yes.)
- Find √2304 by prime factorisation. (Answer: 2304=2⁸×3², √=2⁴×3=48.)
- Generate Pythagorean triplet with m=5. (Answer: 10, 24, 25.)
- √0.0625 = ? (Answer: √(625/10000)=25/100=0.25.)
- How many numbers between 12² and 13²? (Answer: 2×12=24 numbers.)
More Long Division Examples
Example — √7744 by Long Division: Pair: 77 44. Largest square ≤ 77: 8²=64. Remainder = 13. Bring down 44: 1344. Trial divisor = 2×8=16. Find d: (160+d)×d ≤ 1344. d=8: 168×8=1344. √7744 = 88.
Example — √3249: Pair: 32 49. Largest square ≤ 32: 5²=25. Remainder=7. Bring down 49: 749. Trial divisor = 2×5=10. Find d: (100+d)×d ≤ 749. d=7: 107×7=749. √3249 = 57.
Square Roots of Non-Perfect Squares
√2 = 1.41421356... (irrational). You can find approximate values using long division by placing a decimal point in the quotient and adding pairs of zeros. Example — √3 to 3 decimal places: 3.00 00 00. 1²=1. Remainder=2. Bring 00: 200. Trial=2. 27×7=189. Remainder=11. Bring 00: 1100. Trial=34. 343×3=1029. Remainder=71. Bring 00: 7100. Trial=346. 3462×2=6924. Remainder=176. √3 ≈ 1.732.
Applications — Pythagorean Theorem
Find the hypotenuse of a right triangle with legs 5 cm and 12 cm. c² = 5²+12² = 25+144 = 169. c = √169 = 13 cm. 'The 5-12-13 triangle is a Pythagorean triplet. Recognizing triplets saves you from having to calculate square roots.'
Mental Math Tricks
- Numbers ending in 5: 85² = 8×(8+1) followed by 25 = 8×9=72 → 7225.
- To estimate √n: find the nearest perfect square. √50: between 7²=49 and 8²=64. 50−49=1, 64−49=15. Fraction ≈ 1/15 ≈ 0.07. √50 ≈ 7.07.
- Square of numbers near 100: 98² = (100−2)² = 10000−400+4 = 9604. 103² = (100+3)² = 10000+600+9 = 10609.
Finding the Smallest Number to Make a Perfect Square
Example: Find the smallest number by which 720 must be multiplied to get a perfect square. Factorise 720: 720 = 2⁴×3²×5. Check pairs: 2⁴ = (2²)² ← paired. 3² ← paired. 5 ← UNPAIRED. To make 5 pair, multiply by 5. So 720×5 = 3600 = 60². Answer: 5.
Example: Find the smallest number by which 1008 must be divided. 1008 = 2⁴×3²×7. 7 is unpaired. Divide by 7: 1008/7 = 144 = 12². Answer: 7.
Square Roots in Geometry
A square field has area 2025 m². Side = √2025 = 45 m. Perimeter = 4×45 = 180 m. Cost of fencing at ₹15/m = 180×15 = ₹2700.
A right triangle has one leg 9 cm and hypotenuse 15 cm. Other leg = √(15²−9²) = √(225−81) = √144 = 12 cm. 'The Pythagorean theorem creates a natural need for square roots. Most geometry problems end with finding a square root.'
Decimal Square Roots — Common Exam Values
| Number | Square Root |
|---|---|
| 0.01 | 0.1 |
| 0.04 | 0.2 |
| 0.09 | 0.3 |
| 0.16 | 0.4 |
| 0.25 | 0.5 |
| 1.44 | 1.2 |
| 2.25 | 1.5 |
| 6.25 | 2.5 |
'Memorise these. They appear in mensuration and algebra problems where the square root of a decimal is needed.'
