Root Calculator
Square roots, cube roots and any nth root of a number, with the check that raising the answer back gives the original.
The nth root of x is the number that, raised to the power n, gives x: ⁿ√x = x^(1/n).
How the root calculator works
The nth root of x is the number that, raised to the power n, gives x: ⁿ√x = x^(1/n). Even roots of negative numbers are not real; odd roots are, and keep the sign. The calculator also shows the square and cube of the input and whether the input is a perfect square.
Formula: ⁿ√x = x^(1/n)
Worked examples
| Inputs | Root | Note |
|---|---|---|
| √144 | 12 | 12 |
| ∛27 | 3 | 3 |
| ⁵√32 | 2 | 2 |
| √2 | 1.4142135624 | 1.41421356… |
FAQFrequently asked questions
What is the square root of a negative number?
Not a real number; it is an imaginary number (√−4 = 2i). Cube roots of negatives are fine: ∛−8 = −2.
How do I estimate a square root without a calculator?
Find the nearest perfect squares: √50 is between 7 (49) and 8 (64), closer to 7 — about 7.07.
Is the square root always positive?
The principal square root is; the equation x² = 4 has two solutions, 2 and −2.
What is a perfect square?
A whole number that is another whole number squared: 1, 4, 9, 16, 25…
How is the nth root related to exponents?
ⁿ√x = x^(1/n), so roots are just fractional powers.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.