You know √4 = 2 and √9 = 3. But what about √72? Or √0.5? Or figuring out if 196 is a perfect square without a calculator?
Here's how to handle square roots — with and without a calculator.
What Is a Square Root?
The square root of n is the number that, multiplied by itself, gives n.
√n = the number where (number)² = n
- √25 = 5 because 5² = 25
- √144 = 12 because 12² = 144
- √2 ≈ 1.414 because 1.414² ≈ 2
Perfect Squares You Should Memorize
These come up constantly. Memorize at least the first 15:
| n | n² | n | n² | |---|----|---|----| | 1 | 1 | 8 | 64 | | 2 | 4 | 9 | 81 | | 3 | 9 | 10 | 100 | | 4 | 16 | 11 | 121 | | 5 | 25 | 12 | 144 | | 6 | 36 | 13 | 169 | | 7 | 49 | 14 | 196 |
Simplifying Radicals
The key insight: extract the largest perfect square factor.
Example: √72
- Find factors: 72 = 36 × 2
- 36 is a perfect square: √36 = 6
- √72 = 6√2
Example: √50
- 50 = 25 × 2
- √25 = 5
- √50 = 5√2
Example: √48
- 48 = 16 × 3
- √16 = 4
- √48 = 4√3
The Prime Factorization Method
When the largest perfect square factor isn't obvious, use prime factorization:
Example: √180
- 180 = 2 × 2 × 3 × 3 × 5
- Pair up: (2×2) × (3×3) × 5
- Each pair comes out: 2 × 3 = 6 outside
- Unpaired 5 stays inside: √180 = 6√5
Mental Math Shortcuts
| Trick | Example | Method | |-------|---------|--------| | Between perfect squares | √50 is between 7 and 8 | 7²=49, 8²=64 → closer to 7 | | First digit from pairs | √2500 → 25|00 → 50 | Split digits in pairs from right | | Decimal approximation | √5 ≈ 2.24 | 2²=4, 3²=9 → closer to 2 | | ×4 trick | √(4n) = 2√n | √44 = 2√11 |
The Trench Truth: On no-calculator exams, you're usually expected to simplify radicals, not compute decimals. √72 = 6√2 is the "correct" answer. Writing √72 ≈ 8.485 loses points. Always simplify first, then approximate only if asked.
Try our square root calculator to simplify any radical with step-by-step prime factorization.
Related: Quadratic Formula Calculator · Derivative Calculator · Circle Calculator
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