Root Calculator
nth root of a number, plus square and cube roots.
3th root of 64
4
nth root
Square root
8
Cube root
4
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Root Calculator works
The GKCalculators Root Calculator computes the nth root of any number, including the common special cases of square roots and cube roots, delivering instant, precise results. It's built for students, engineers, and anyone who needs to work with radical expressions without reaching for a scientific calculator.
At its core, this calculator finds the value that, when multiplied by itself a specified number of times, equals your input. For a square root, it finds the number you multiply by itself twice; for a cube root, three times; and for a general nth root, n times. Internally, it uses floating-point exponentiation — computing x^(1/n) — which is mathematically equivalent to the radical notation ⁿ√x and gives results accurate to many decimal places for virtually any input you're likely to encounter.
The two most common operations are square roots and cube roots, which deserve special attention. Square roots are only real-valued for non-negative inputs in the real number system, which is why entering a negative number under an even-indexed radical produces a complex result (the calculator flags this clearly). Cube roots, by contrast, are defined for all real numbers, including negatives — the cube root of -27 is -3 — because multiplying a negative number by itself three times yields a negative product. Understanding this distinction prevents one of the most common errors users make when working with radicals.
For the general nth root, the parity of n determines the rules. Even roots (4th, 6th, 8th, etc.) of negative numbers are not real, while odd roots (5th, 7th, 9th, etc.) of negative numbers are real and negative. Fractional or decimal values of n are also supported: entering n = 0.5 is identical to a square root, and n = 1/3 is a cube root. This flexibility makes the calculator useful for problems in exponential growth, physics (e.g., finding the radius from a volume), finance (e.g., CAGR calculations), and dimensional analysis.
A common mistake is confusing the index and the radicand. The index is n (the small number indicating which root you want), and the radicand is the number under the radical sign. Another pitfall is expecting whole-number answers: most roots are irrational — they go on forever without repeating. For example, √2 ≈ 1.41421356…, and no finite decimal is exact. The calculator returns a high-precision decimal approximation, which is appropriate for real-world use. If you need an exact simplified radical form (like 3√5), that requires a separate simplification step, which is a good complement to the numerical answer this tool provides.
Formula
ⁿ√x = x^(1/n)
Pro tips
- To verify your answer, raise the result to the power of n and check that it equals your original number — for example, if ³√64 = 4, then 4³ should equal 64.
- For cube roots of negative numbers, enter the negative sign as part of the radicand and select index 3; this is valid and will return a negative result, unlike even-index roots of negatives.
- Use the nth root function with n = 1/2 or n = 0.5 as a quick way to cross-check a square root result, since x^(1/2) and √x are mathematically identical.
- In finance, use the nth root to calculate Compound Annual Growth Rate (CAGR): take the nth root of (End Value / Start Value) where n is the number of years, then subtract 1.
- If you need a simplified radical form rather than a decimal, first check whether the radicand has any perfect-power factors matching your index — for instance, √72 = √(36×2) = 6√2 — then use this calculator for the decimal confirmation.
Key terms
- Radical
- — The symbol (√) used to denote a root operation, where the expression under the symbol is called the radicand.
- Radicand
- — The number or expression placed under the radical sign, whose root you are computing.
- Index (n)
- — The small number written above and to the left of the radical symbol that specifies which root to take — 2 for square root, 3 for cube root, and so on.
- Square Root
- — The 2nd root of a number x, written √x, which returns the non-negative value that, when squared, equals x.
- Cube Root
- — The 3rd root of a number x, written ³√x, which returns the value that, when cubed, equals x — defined for all real numbers including negatives.
- Principal Root
- — The conventional, non-negative real root returned for positive radicands when multiple roots technically exist, such as both +3 and -3 being square roots of 9.
