Square, cube, and nth roots — instant and exact
Root Calculator
Result
Both +8 and -8 satisfy x² = 64
Common Roots Reference
| Number | √ (n=2) | ∛ (n=3) | ⁴√ (n=4) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 4 | 2 | ≈1.587 | ≈1.414 |
| 8 | ≈2.828 | 2 | ≈1.682 |
| 16 | 4 | ≈2.520 | 2 |
| 25 | 5 | ≈2.924 | ≈2.236 |
| 27 | ≈5.196 | 3 | ≈2.280 |
| 64 | 8 | 4 | ≈2.828 |
| 100 | 10 | ≈4.642 | ≈3.162 |
| 125 | ≈11.18 | 5 | ≈3.344 |
| 256 | 16 | ≈6.350 | 4 |
How Roots Work
A root calculator lets you find the square root, cube root, or any nth root of any real number in seconds. Whether you need to simplify a radical expression, check a perfect power, or compute a fractional exponent for engineering or school, this free tool handles it all — showing the principal root, the negative counterpart for even roots, and whether the result is exact or irrational.
A square root (n=2) finds the value y where y² = x. A cube root (n=3) finds y where y³ = x. The key practical difference: square roots only exist as real numbers for non-negative inputs, while cube roots are defined for all real numbers including negatives — for example ∛(−8) = −2.
Squaring any real number — positive or negative — always produces a non-negative result. So no real number y satisfies y² = −4, for instance. The result exists only in the complex number system (as 2i). Odd roots like cube roots do work on negatives because cubing a negative gives a negative.
A perfect square is an integer whose square root is also an integer (e.g. 64 = 8²). A perfect cube has an integer cube root (e.g. 27 = 3³). More generally, a perfect nth power has an integer nth root. The calculator flags these results as 'Exact'.
Select 'Nth Root' and enter 1.5 as the index. The nth root of x equals x^(1/n), so the 1.5th root is x^(1/1.5) = x^(2/3). Fractional and decimal root indices are fully supported.