source: ncert
topic: numbers
class_level: class_8
chapter: ch6
difficulty: beginner
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.