Squares and Square Roots

12

Step 1: We need to find the positive square root of 144. Step 2: Recall that finding a square root is the inverse of squaring. Step 3: We check: 12 × 12 = 144. Step 4: Since 12² = 144, we conclude √144 = 12. Step 5: Verify: 11² = 121 (too small), 13² = 169 (too big), so 12 is the only correct answer.

25

Step 1: The first 5 odd natural numbers are 1, 3, 5, 7, 9. Step 2: Their sum = 1 + 3 + 5 + 7 + 9. Step 3: 1+3=4, 4+5=9, 9+7=16, 16+9=25. Step 4: The sum is 25. Step 5: This follows the rule that the sum of the first n odd natural numbers = n². Here n=5, so the sum = 5² = 25.

438

Step 1: Perfect squares can only end in the digits 0, 1, 4, 5, 6, or 9. Step 2: 361 ends in 1 — possible perfect square. Step 3: 529 ends in 9 — possible perfect square. Step 4: 438 ends in 8 — a perfect square CANNOT end in 2, 3, 7, or 8. So 438 cannot be a perfect square. Step 5: 625 ends in 5 — possible perfect square. So 438 is definitely NOT a perfect square.

16

Step 1: Find prime factorisation of 256. Step 2: 256 = 2×128 = 2×2×64 = 2×2×2×32 = 2×2×2×2×16 = 2×2×2×2×2×8 = 2×2×2×2×2×2×4 = 2×2×2×2×2×2×2×2 = 2⁸. Step 3: Pair the prime factors: (2×2)×(2×2)×(2×2)×(2×2) = 4 pairs. Step 4: √256 = 2×2×2×2 = 16. Step 5: Verify: 16×16 = 256. ✓