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√ Square Root Calculator
Enter any positive number to find its square root. Includes decimal result, exact radical form, and step-by-step Babylonian method.
Quick Examples
Perfect Squares Quick Check
Enter a number and click Calculate
Decimal result, exact radical form, perfect square check, and step-by-step Babylonian working will appear here instantly.
How It Works
Square Root Methods — Babylonian Algorithm, Simplification & Radical Form
Most calculators just give you the decimal. Understanding how to find and simplify square roots is what students and engineers actually need — and what most online tools skip entirely.
The Babylonian (Newton's) Method
Iterative Square Root Algorithm
x_new = (x + n/x) ÷ 2
Start with an initial guess x (try n/2 or 1)
Each iteration doubles correct decimal places
Stop when |x_new − x| < desired precision
Example: √2 by Babylonian Method
1
Initial guess: x = 1
2
x = (1 + 2/1) ÷ 2 = 1.5
3
x = (1.5 + 2/1.5) ÷ 2 = 1.41667
4
x = (1.41667 + 2/1.41667) ÷ 2 = 1.41422
✓ √2 ≈ 1.41421356 — converges in just 4 iterations
Simplifying Square Roots (Radical Form)
Simplification Method
√n = √(a² × b) = a√b
Find the largest perfect square factor of n
a = √(perfect square factor), b = remaining factor
Example: √72 = √(36×2) = 6√2
Prime factor method: 72 = 2³×3² → pairs: 2¹×3 = 6, remaining: 2 → 6√2
Cube Roots & Nth Roots
General Root Formula
ⁿ√x = x^(1/n)
Square root: n=2 → √x = x^0.5
Cube root: n=3 → ∛x = x^(1/3)
4th root: n=4 → x^0.25
Odd roots of negatives are real: ∛(−8) = −2
Even roots of negatives are imaginary: √(−4) = 2i
Negative Numbers & Imaginary Results
Imaginary Unit i
√(−n) = i√n
i = √(−1) — the imaginary unit
√(−4) = 2i because (2i)² = 4i² = 4×(−1) = −4 ✓
√(−50) = 5i√2 (simplify √50 = 5√2 first)
Used in complex numbers, AC circuits, quantum mechanics
Is a Number Rational or Irrational?
- √n is rational only if n is a perfect square (1, 4, 9, 16, 25…)
- √n is irrational for all other positive integers — non-repeating, non-terminating decimal
- √2 = 1.41421356… was proven irrational by ancient Greeks — a landmark in math history
- Irrational square roots cannot be expressed as fractions
Reference Table
Square Root Table — Numbers 1 to 200 (2026)
Quick reference for square roots 1 through 200, including simplified radical form where applicable. Use the calculator above for any value not listed.
| n | √n (decimal) | Simplified Radical | Perfect Square? |
|---|---|---|---|
| 1 | 1.000000 | 1 | ✅ Yes |
| 2 | 1.414214 | √2 | — |
| 3 | 1.732051 | √3 | — |
| 4 | 2.000000 | 2 | ✅ Yes |
| 5 | 2.236068 | √5 | — |
| 6 | 2.449490 | √6 | — |
| 7 | 2.645751 | √7 | — |
| 8 | 2.828427 | 2√2 | — |
| 9 | 3.000000 | 3 | ✅ Yes |
| 10 | 3.162278 | √10 | — |
| 11 | 3.316625 | √11 | — |
| 12 | 3.464102 | 2√3 | — |
| 13 | 3.605551 | √13 | — |
| 14 | 3.741657 | √14 | — |
| 15 | 3.872983 | √15 | — |
| 16 | 4.000000 | 4 | ✅ Yes |
| 17 | 4.123106 | √17 | — |
| 18 | 4.242641 | 3√2 | — |
| 19 | 4.358899 | √19 | — |
| 20 | 4.472136 | 2√5 | — |
| 24 | 4.898979 | 2√6 | — |
| 25 | 5.000000 | 5 | ✅ Yes |
| 27 | 5.196152 | 3√3 | — |
| 30 | 5.477226 | √30 | — |
| 32 | 5.656854 | 4√2 | — |
| 36 | 6.000000 | 6 | ✅ Yes |
| 40 | 6.324555 | 2√10 | — |
| 45 | 6.708204 | 3√5 | — |
| 48 | 6.928203 | 4√3 | — |
| 49 | 7.000000 | 7 | ✅ Yes |
| 50 | 7.071068 | 5√2 | — |
| 54 | 7.348469 | 3√6 | — |
| 60 | 7.745967 | 2√15 | — |
| 63 | 7.937254 | 3√7 | — |
| 64 | 8.000000 | 8 | ✅ Yes |
| 72 | 8.485281 | 6√2 | — |
| 75 | 8.660254 | 5√3 | — |
| 80 | 8.944272 | 4√5 | — |
| 81 | 9.000000 | 9 | ✅ Yes |
| 90 | 9.486833 | 3√10 | — |
| 96 | 9.797959 | 4√6 | — |
| 98 | 9.899495 | 7√2 | — |
| 100 | 10.000000 | 10 | ✅ Yes |
| 108 | 10.392305 | 6√3 | — |
| 120 | 10.954451 | 2√30 | — |
| 121 | 11.000000 | 11 | ✅ Yes |
| 125 | 11.180340 | 5√5 | — |
| 128 | 11.313708 | 8√2 | — |
| 144 | 12.000000 | 12 | ✅ Yes |
| 147 | 12.124356 | 7√3 | — |
| 150 | 12.247449 | 5√6 | — |
| 160 | 12.649111 | 4√10 | — |
| 162 | 12.727922 | 9√2 | — |
| 169 | 13.000000 | 13 | ✅ Yes |
| 175 | 13.228757 | 5√7 | — |
| 180 | 13.416408 | 6√5 | — |
| 192 | 13.856406 | 8√3 | — |
| 196 | 14.000000 | 14 | ✅ Yes |
| 200 | 14.142136 | 10√2 | — |
Perfect Squares 1 to 400
| n (integer) | n² (perfect square) | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
| 16 | 256 | 16 |
| 17 | 289 | 17 |
| 18 | 324 | 18 |
| 19 | 361 | 19 |
| 20 | 400 | 20 |
Simplified Radical Form — Common Examples
| Original | Prime Factorisation | Simplified Form | Decimal |
|---|---|---|---|
| √8 | 8 = 2³ | 2√2 | 2.828427 |
| √12 | 12 = 2²×3 | 2√3 | 3.464102 |
| √18 | 18 = 2×3² | 3√2 | 4.242641 |
| √20 | 20 = 2²×5 | 2√5 | 4.472136 |
| √24 | 24 = 2³×3 | 2√6 | 4.898979 |
| √27 | 27 = 3³ | 3√3 | 5.196152 |
| √32 | 32 = 2⁵ | 4√2 | 5.656854 |
| √45 | 45 = 3²×5 | 3√5 | 6.708204 |
| √48 | 48 = 2⁴×3 | 4√3 | 6.928203 |
| √50 | 50 = 2×5² | 5√2 | 7.071068 |
| √54 | 54 = 2×3³ | 3√6 | 7.348469 |
| √72 | 72 = 2³×3² | 6√2 | 8.485281 |
| √75 | 75 = 3×5² | 5√3 | 8.660254 |
| √80 | 80 = 2⁴×5 | 4√5 | 8.944272 |
| √98 | 98 = 2×7² | 7√2 | 9.899495 |
| √108 | 108 = 2²×3³ | 6√3 | 10.392305 |
| √125 | 125 = 5³ | 5√5 | 11.180340 |
| √147 | 147 = 3×7² | 7√3 | 12.124356 |
| √162 | 162 = 2×3⁴ | 9√2 | 12.727922 |
| √200 | 200 = 2³×5² | 10√2 | 14.142136 |
FAQ
Square Root Calculator — Frequently Asked Questions
Everything you need to know about calculating, simplifying, and understanding square roots, cube roots, and imaginary numbers.