Core answer: √x asks "what non-negative number squared gives x": √144 = 12 because 12² = 144. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) come out whole; everything else is irrational and needs estimation or a calculator: √2 ≈ 1.414, √3 ≈ 1.732, √10 ≈ 3.162.
Estimating any root in seconds
Bracket between perfect squares, then refine: √50 sits between 49 (7²) and 64 (8²), and 50 is 1/15 of the way up → ≈ 7.07. (Actual: 7.0711.)
The Newton–Babylon iteration converges absurdly fast: to find √S, start with guess g, then repeat g' = (g + S/g)/2. For √50 with g = 7: g' = (7 + 50/7)/2 = 7.071 — one step to three decimals.
Where square roots appear
| Context | Formula | Example |
|---|---|---|
| Diagonal of a square | d = a√2 | 10 cm tile → 14.14 cm diagonal |
| TV/monitor size | diagonal = √(w²+h²) | 121×68 cm screen → 55 in |
| Pythagoras | c = √(a²+b²) | 3-4-5 right triangle |
| Standard deviation | √(mean of squared deviations) | statistics |
| Free fall time | t = √(2h/g) | 20 m drop → 2.02 s |
| RMS voltage | 220 V RMS = 311 V peak | household power |
Example — the 3-4-5 rule. Builders square a corner by measuring 3 m and 4 m along two walls; the diagonal must read √(9+16) = 5 m, or the angle is not 90°.
Properties that save work
- √(ab) = √a × √b → √50 = √25 × √2 = 5√2 (simplified radical form)
- √(a/b) = √a / √b
- √a + √b does NOT simplify — √9 + √16 = 7 ≠ √25 = 5. The single most common algebra error.
Common mistakes and myths
- √(a+b) = √a + √b — false; √(9+16) = 5, not 7.
- "√x = ±" — the √ symbol denotes the principal (non-negative) root only; ± appears when *solving* x² = a.
- Roots of negatives — √(−9) is not −3; it is 3i (imaginary), undefined in the reals.
- Forgetting units scale — doubling area multiplies side by √2 ≈ 1.41, not 2; a "double" pizza (2× area) is only 1.41× the diameter.
- Believing all roots are messy — √(2.25) = 1.5 exactly; decimals can be perfect squares too.