Squares and Square Roots

69

Step 1: Group digits from right: 47 | 61. Step 2: Find largest number whose square ≤ 47. Since 6² = 36 ≤ 47 and 7² = 49 > 47, write 6 as quotient. Subtract 36 from 47, remainder = 11. Step 3: Bring down 61, new dividend = 1161. Double quotient: 2 × 6 = 12. Find digit d such that 12d × d ≤ 1161. Try d = 9: 129 × 9 = 1161. Perfect! Step 4: Subtract 1161 − 1161 = 0. Quotient = 69. Verify: 69 × 69 = 4761. ✓

0.035

Step 1: Write 0.001225 as a fraction: 1225/1000000. Step 2: Find √1225 and √1000000 separately. Step 3: 1225 = 5 × 5 × 7 × 7 = (5×7)² = 35², so √1225 = 35. Step 4: √1000000 = √(10^6) = 10^3 = 1000. Step 5: √(1225/1000000) = 35/1000 = 0.035. Verify: (0.035)² = 0.001225. ✓ Common mistake: Students often write 0.35 by forgetting the correct number of decimal places.

8000

Step 1: A perfect square ends in 0, 1, 4, 5, 6, or 9. Also, a number ending in an odd number of zeroes can NEVER be a perfect square. Step 2: 8000 ends in 3 zeroes (odd number of zeroes), so it cannot be a perfect square. Step 3: Check others — 1024 = 32², 2025 = 45², 4900 = 70² (ends in 2 zeroes which is even). Step 4: 8000 = 2^6 × 5^3 — the factor 5 appears 3 times (odd), so it cannot be a perfect square. Answer: 8000.

1(2/3)

Step 1: Convert mixed fraction to improper fraction: 2(7/9) = (18+7)/9 = 25/9. Step 2: Apply rule: √(a/b) = √a / √b. So √(25/9) = √25 / √9. Step 3: √25 = 5 and √9 = 3. Step 4: √(25/9) = 5/3 = 1(2/3). Verify: (5/3)² = 25/9 = 2(7/9). ✓ Common mistake: Students sometimes try to find √2 and √(7/9) separately, which is incorrect.