What is the square root of 81?
√81 = 9 because 9 × 9 = 81.
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Principal square root √a for geometry, radical simplification, and converting variance back to standard deviation units.
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The square root of a non-negative number a is the value x such that x² = a. √81 = 9 because 9×9 = 81. Every positive number has two square roots; the calculator shows the principal (non-negative) root.
Square-root outputs are numerical approximations where needed; use domain-specific precision and rounding standards where required.
Square root undoes squaring. A square with area 50 m² has side length √50 ≈ 7.07 m—geometry constantly asks which number multiplied by itself gives the area.
**Perfect squares**—1, 4, 9, 16, 25, 36, 49, 64, 81, 100—have integer roots. **Imperfect squares** like 2 or 50 produce irrational decimals that never terminate.
**Principal root:** √a ≥ 0 for a ≥ 0. Equation x² = 9 has solutions x = 3 and x = −3, but √9 = 3 by convention.
**Connection to exponents:** √a = a^(1/2). **Connection to statistics:** population standard deviation is the square root of variance—spread measured in original units after squaring deviations.
Simplify radicals by factoring out perfect squares: √72 = √(36×2) = 6√2.
Enter a non-negative number for √a—principal root with optional simplification context for perfect-square factors.
Concise answers for common searches — definitions, steps, and comparisons.
√81 = 9 because 9 × 9 = 81.
√50 = √(25×2) = 5√2 ≈ 7.071.
No. Negative radicands have no real square root—the principal root is defined for a ≥ 0.
σ = √(variance). If variance is 16, standard deviation is 4 in original data units.
Uses IEEE double-precision square root for non-negative inputs. Validates a ≥ 0 for real results.
Formula
If x² = a and a ≥ 0, principal root x = √a = a^(1/2). Distance formula: d = √((x₂−x₁)² + (y₂−y₁)²). σ = √(Σ(x−μ)²/N).Type the radicand—area, squared sum, or variance before root.
Decimal result; for perfect squares you get an integer.
Split a into largest square factor × remainder for simplified radical form on paper.
a^0.5 should match √a—cross-check both tools.
Input
√144Output
1212 × 12 = 144.
Input
√(3² + 4²)Output
5Classic 3-4-5 triangle hypotenuse.
Input
√2Output
≈ 1.414214Not a fraction—decimal continues without repeating.
Input
√72Output
≈ 8.485; exact form 6√272 = 36×2 → √36 × √2 = 6√2.
Input
√0, √1Output
0 and 1Boundary cases: 0² = 0, 1² = 1.
Input
Variance = 49Output
σ = 7Standard deviation restores original measurement units.
Common real-world scenarios where this tool saves time.
Right triangle legs 5 and 12: hypotenuse = √(5² + 12²) = √169 = 13.
Square garden area 200 ft² → side ≈ √200 ≈ 14.14 ft.
Variance 25 (units²) → σ = √25 = 5 in original units.
Confirm √50 = 5√2 after factoring 25×2.
Step-by-step chains that connect related tools for common tasks.
| √a | a | √a (integer) |
|---|---|---|
| √4 | 4 | 2 |
| √25 | 25 | 5 |
| √49 | 49 | 7 |
| √100 | 100 | 10 |
| √169 | 169 | 13 |
| √400 | 400 | 20 |
| √a | Decimal (6 dp) | Simplified form |
|---|---|---|
| √2 | 1.414214 | — |
| √3 | 1.732051 | — |
| √5 | 2.236068 | — |
| √50 | 7.071068 | 5√2 |
| √18 | 4.242641 | 3√2 |
400 cap catches most factorization shortcuts in radical simplification.
Variance output squared units—square root returns to data units.
Multiply result by itself—should recover input within float tolerance.
Algebra courses interchange √a and a^(1/2)—same numeric answer.
√ symbol means principal root only. Equation x² = 25 has solutions ±5; √25 = 5.
√(9 + 16) = 5 but √9 + √16 = 7. Root does not distribute over addition.
Area and lengths are non-negative—negative radicands need complex analysis.
Keep √(sum of squares) exact until final answer—avoid √(12.5) rounding mid-problem.
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√x undoes squaring for non-negative x. (√x)² = x, but √(x²) = |x|.
c = √(a² + b²) for legs a and b in a right triangle.
No—√2 is irrational. Its decimal neither terminates nor repeats.
0 × 0 = 0—the only square root of zero is zero.
Both invert growth operations: log inverts exp; squaring inverts principal square root.
Cube root x where x³ = a—use exponent 1/3, not this square root tool.
When a ≥ 0 and b ≥ 0. Product rule fails if either factor is negative in reals.
Radicand values are not uploaded—computed locally.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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For powers (bn), use the Exponent calculator.
Tip: odd degrees allow negative inputs.
Expression
√(144)
Result
12
Formula: √(x) = x^(1/2)
Perfect square detected
144 = 12²
Therefore: √144 = 12
Exact (simplified radical)
√144 = 12
Shareable link: https://everydaytools.pro/square-root-calculator?value=144°ree=2&decimals=8
All calculations run locally in your browser — no data stored.