Square roots by factorisation and division

Learning outcome

Find roots by factorisation and digit-pair division; distinguish exact roots, integer remainders and rounded decimals.

Concepts and assumptions

For a ≥ 0, √a is the nonnegative principal root. For a > 0, x² = a instead has solutions ±√a.

For positive integer perfect squares, take one prime from each factor pair to obtain the exact root; squaring reverses this.

Pair digits outward from the decimal point: integers leftward, decimals rightward. Allow a single-digit first integer group; pad an incomplete decimal pair with a right-hand zero.

Initially q = R = 0. Bring down each pair b: T = 100 × R + b. Choose the largest digit d from 0–9 with (20 × q + d) × d ≤ T, set R = T minus that product, and append d to q.

This works because (10 × q + d)² - (10 × q)² = (20 × q + d) × d.

Place the root’s decimal point after integer groups; q ignores that point. Append 00 pairs to continue.

For integer N ≥ 0, its integer square root k satisfies k² ≤ N < (k + 1)²; its remainder is N - k². Decimal-stage remainders are scaled, not part of √N. For two-decimal rounding, find a third digit; 5 or greater rounds upward.

Worked examples

Example 1 — Factorisation. Since 7056 = 16 × 441 = 2⁴ × 3² × 7², √7056 = 2² × 3 × 7 = 84. Check: 84² = 7056.

Example 2 — Preserve a zero digit. For √10404, use pairs 1 | 04 | 04. First, 1² = 1; root = 1, remainder = 0. Bring down 04: T = 4. Digit 0 gives (20 + 0) × 0 = 0; digit 1 would give 21 > 4. Root = 10, remainder = 4. Bring down 04: T = 404. Choose 2: (200 + 2) × 2 = 404. Root = 102, remainder = 0, so √10404 = 102 exactly.

Example 3 — Decimal approximation. For √27, 5² = 25 gives integer root 5, remainder 2. Append 00: T = 200; choose 1 since 101 × 1 = 101, leaving 99; root digits = 51. Append 00: T = 9900; choose 9 since 1029 × 9 = 9261, leaving 639; digits = 519. Append 00: T = 63900; choose 6 since 10386 × 6 = 62316, leaving 1584; digits = 5196. Thus 5.196 < √27 < 5.197, giving √27 ≈ 5.20 to two decimal places.

Common mistakes

Do not drop zero root digits, mispair decimals or write an approximation with an equality sign.

Practice questions

1. Find √2025 by prime factorisation.
2. Find √4624 by digit-pair division.
3. Find the integer square root and remainder of 731.
4. Find √0.0625 using digit pairs.

Worked solutions

1. 2025 = 81 × 25 = 3⁴ × 5². Therefore √2025 = 3² × 5 = 45.
2. Pairs: 46 | 24. Choose 6: 46 - 36 = 10. Bring down 24: T = 1024. Choose 8: (120 + 8) × 8 = 1024, leaving 0. Therefore √4624 = 68.
3. Pairs: 7 | 31. Choose 2: 7 - 4 = 3. Bring down 31: T = 331. Choose 7: 47 × 7 = 329, leaving 2. Thus k = 27; 731 = 27² + 2 < 28².
4. Pairs: 0 . 06 | 25. Integer root digit = 0. Bring down 06: 2 × 2 = 4 leaves 2. Bring down 25: T = 225; (40 + 5) × 5 = 225 leaves 0. Thus √0.0625 = 0.25.