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Compute the principal square root, cube root, square, and cube of any real number, with support for negative cube roots and irrational results like √2 for Pythagorean theorem work.
Math
Generated on May 23, 2026
Compute the principal square root, cube root, square, and cube of any real number, with support for negative cube roots and irrational results like √2 for Pythagorean theorem work.
A square-root calculator computes the principal square root, cube root, square, and cube of any real number. Roots are everywhere in practical math: the distance formula uses a square root, standard deviation uses a square root, physics equations for kinetic energy and speed involve square roots, and construction applies the Pythagorean theorem constantly. This tool handles both positive and negative inputs appropriately — cube roots of negatives are real, while square roots of negatives have no real solution (only imaginary).
Formula
√n = n^(1/2). ∛n = n^(1/3). x² = x × x. x³ = x × x × x.Pythagorean theorem, the classic: a right triangle with legs 3 and 4 has a hypotenuse of √(3² + 4²) = √25 = 5. You've used this since middle school whether you remember or not — finding the diagonal of a room to fit a couch, calculating the shortest path across a park, computing distance between two GPS coordinates. Square roots aren't abstract; they're how you convert a squared quantity (area, energy, variance) back into a linear measure. This calculator handles principal square roots, cube roots (including of negatives, which is fine for odd-degree roots), squares, and cubes — the four operations that cover 95 percent of practical needs. Note one important asymmetry: √(−9) is rejected because the square root of a negative is imaginary (3i, not a real number), but ∛(−8) = −2 is perfectly valid because the cube root of a negative just gives you a negative real. Odd-degree roots are friendly to negatives; even-degree roots aren't. Beyond the Pythagorean theorem, square roots show up in standard deviation (the 'standard' in standard deviation literally means 'square root of variance'), kinetic energy formulas, GPS distance calculations, and even Black-Scholes option pricing in finance. Worth respecting as one of the half-dozen most-used operations in applied math.
A square-root calculator computes the principal square root, cube root, square, and cube of any real number. Roots are everywhere in practical math: the distance formula uses a square root, standard deviation uses a square root, physics equations for kinetic energy and speed involve square roots, and construction applies the Pythagorean theorem constantly. This tool handles both positive and negative inputs appropriately — cube roots of negatives are real, while square roots of negatives have no real solution (only imaginary).
The square root is the inverse of squaring — it finds the number that, multiplied by itself, produces n. Every positive real number has two square roots (one positive, one negative); the 'principal' square root returns the positive one by convention. Cube root is the inverse of cubing, and because odd-degree functions preserve sign, cube roots of negative numbers are real and negative — ∛(−27) = −3 because (−3)³ = −27.
Square roots and cube roots of common numbers. Perfect squares and cubes are marked; others are rounded to 4 decimals.
| n | √n | ∛n | n² | n³ |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1.4142 | 1.2599 | 4 | 8 |
| 3 | 1.7321 | 1.4422 | 9 | 27 |
| 4 | 2 ✓ | 1.5874 | 16 | 64 |
| 5 | 2.2361 | 1.7100 | 25 | 125 |
| 8 | 2.8284 | 2 ✓ | 64 | 512 |
| 9 | 3 ✓ | 2.0801 | 81 | 729 |
| 10 | 3.1623 | 2.1544 | 100 | 1,000 |
| 16 | 4 ✓ | 2.5198 | 256 | 4,096 |
| 25 | 5 ✓ | 2.9240 | 625 | 15,625 |
| 27 | 5.1962 | 3 ✓ | 729 | 19,683 |
| 36 | 6 ✓ | 3.3019 | 1,296 | 46,656 |
| 49 | 7 ✓ | 3.6593 | 2,401 | 117,649 |
| 64 | 8 ✓ | 4 ✓ | 4,096 | 262,144 |
| 100 | 10 ✓ | 4.6416 | 10,000 | 1,000,000 |
√2 ≈ 1.41421356 — the diagonal of a unit square; also the ratio of paper sizes (A4 is A5's diagonal ratio).
√3 ≈ 1.73205 — the diagonal of a unit cube in 3D space; used in electrical engineering (line-to-phase voltage).
√π ≈ 1.7725 — appears in Gaussian probability distributions.
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 — first 10 integer squares.
√64 = 8, because 8² = 64 — a clean perfect square.
∛27 = 3, because 3³ = 27 — a clean perfect cube.
√2 ≈ 1.41421356… — irrational, used in the Pythagorean theorem for a 1-1 right triangle and in paper sizing (A4 dimensions are a factor of √2).
√(−9) has no real value (imaginary is 3i) — this calculator returns an error for negative square-root input.
∛(−8) = −2 — cube roots of negatives are real and negative.
The Pythagorean theorem: a right triangle with legs 3 and 4 has hypotenuse √(3² + 4²) = √25 = 5.
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