| --- |
| source: ncert |
| topic: numbers |
| class_level: class_8 |
| chapter: ch6 |
| difficulty: beginner |
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| |
| Class 8 | Ch6: Squares and Square Roots |
| PERFECT SQUARES: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225... |
| Properties: |
| A number ending in 2,3,7,8 is NEVER a perfect square. |
| A perfect square ending in 1 → number ends in 1 or 9. |
| A perfect square ending in 4 → number ends in 2 or 8. |
| A perfect square ending in 9 → number ends in 3 or 7. |
| A perfect square ending in 6 → number ends in 4 or 6. |
| Number of zeros at end of perfect square is always EVEN. |
| SQUARE ROOT METHODS: |
| Method 1 - Prime factorisation: pair up prime factors, take one from each pair. |
| sqrt(1764): 1764=2^2×3^2×7^2 → sqrt=2×3×7=42. |
| Method 2 - Long division: group digits in pairs from right, find largest square≤group. |
| PYTHAGOREAN TRIPLETS PATTERN: |
| For any m>1: (2m, m²-1, m²+1) form a Pythagorean triplet. |
| m=2: (4,3,5). m=3: (6,8,10). m=4: (8,15,17). |
| ESTIMATING SQUARE ROOT: find between which two consecutive perfect squares the number lies. |
| sqrt(200): between sqrt(196)=14 and sqrt(225)=15. Closer to 14. |
| SOLVED EXAMPLES: |
| Example 1: Is 1352 a perfect square? |
| 1352=2^3×13^2. Power of 2 is odd (3), so NOT a perfect square. |
| To make perfect square: multiply by 2 → 2704=2^4×13^2=(2^2×13)^2=52^2. |
| Example 2: Find least number to multiply 675 to make perfect square. |
| 675=3^3×5^2. Need one more 3: multiply by 3 → 2025=45^2. |
| COMMON MISTAKES: |
| sqrt(a^2+b^2) ≠ a+b. sqrt(25+144)=sqrt(169)=13, NOT 5+12=17. |
| A number ending in 0 is perfect square only if it ends in EVEN number of zeros. |
