Calculate any root — 4th, 5th, or any custom index — instantly. Works for square roots, cube roots, and beyond. Get exact results with simplified forms. 100% free.
By now you might already know square roots (racine carrée) and cube roots (racine cubique) — but what happens when you need the 4th root, the 5th root, or any root beyond those two? That's exactly what the nth root calculator above is built for.
An nth root generalizes the idea of square roots and cube roots to work for any number of multiplications, not just two or three. The nth root of a number asks, "what value, multiplied by itself n times, gives me this number?"
Example: The 4th root of 16 is 2, because 2 × 2 × 2 × 2 = 16. Once you see that square roots and cube roots are just two specific cases of this same idea, the nth root stops feeling abstract.
Enter your number (the radicand) into the first box.
Enter the index — the "n" in nth root — into the second box (for example, enter 4 for a 4th root, or 5 for a 5th root).
Click Calculate to get your answer instantly, including a simplified form when one's available.
The nth root uses the same radical symbol (√) as square roots, but with a small number tucked into the notch to show the index:
When no index is shown (just √x), it's assumed to mean n = 2, which is why square roots don't need a visible "2."
You'll also see nth roots written using fractional exponents:
Both notations mean exactly the same thing.
Just like square and cube roots, the nth root is defined by a relationship rather than a plug-in formula:
In plain words: whatever answer you get, raising it to the power of n has to bring you back to your original number.
It's worth pausing on these two terms, because they show up constantly:
2 for square root, 3 for cube root, 4 for 4th root, and so on.
For example, in ⁵√32: 5 is the index, 32 is the radicand, and the answer is 2 (because 2⁵ = 32).
Seeing several indices side by side makes the pattern click faster than any single example can.
| Root | Expression | Answer | Why |
|---|---|---|---|
| Square root (n=2) | √64 | 8 | 8 × 8 = 64 |
| Cube root (n=3) | ∛64 | 4 | 4 × 4 × 4 = 64 |
| 4th root (n=4) | ⁴√81 | 3 | 3 × 3 × 3 × 3 = 81 |
| 5th root (n=5) | ⁵√243 | 3 | 3 × 3 × 3 × 3 × 3 = 243 |
| 6th root (n=6) | ⁶√729 | 3 | 3⁶ = 729 |
This is the single most important rule that makes nth roots different from square and cube roots on their own — because it depends entirely on whether your index is even or odd.
Example: ⁵√(−32) = −2, because (−2) × (−2) × (−2) × (−2) × (−2) = −32
Example: ⁴√(−16) has no real solution, because no real number multiplied by itself four times can produce a negative result.
Multiplying a negative number by itself an even number of times always lands on positive, while multiplying it an odd number of times keeps the negative sign. This one rule explains why cube roots (odd index) handle negatives just fine, while square roots and 4th roots (even indexes) don't.
Just like square root simplification and cube root simplification, nth roots can often be simplified by pulling out perfect-power factors.
The same logic that works for square and cube roots extends cleanly to any index — just look for the largest matching perfect power hiding inside your radicand.
Nth roots build on everything you know from square roots and cube roots, extending those concepts into fields that require higher-order calculations:
Compound Annual Growth Rate (CAGR) is calculated using an nth root: CAGR = ⁿ√(End/Start) − 1. For a 5-year investment, that's a 5th root. For 10 years, a 10th root. This is the extension of the 3-year cube root method to any time horizon.
The geometric mean of n data points uses the nth root: ⁿ√(x₁·x₂·...xₙ). It's preferred over the arithmetic mean for growth rates, ratios, and normalized scores — anywhere multiplicative effects matter more than additive ones.
Higher-dimensional scaling problems in aerodynamics, materials science, and signal processing use 4th, 5th, and higher roots. When a physical quantity depends on a variable raised to the nth power, the inverse operation requires the nth root.
RSA encryption uses modular exponentiation where the nth root operation (modular root extraction) is computationally hard — and that hardness is what makes the encryption secure. Algorithm complexity analysis also involves nth roots for certain bounds.
A quick-reference table connecting radicands, indexes, and their exact whole-number roots.
| Expression | Radicand | Index | Answer |
|---|---|---|---|
| ⁴√16 | 16 | 4 | 2 |
| ⁵√32 | 32 | 5 | 2 |
| ⁶√64 | 64 | 6 | 2 |
| ⁴√81 | 81 | 4 | 3 |
| ⁵√243 | 243 | 5 | 3 |
| ⁶√729 | 729 | 6 | 3 |
| ³√64 | 64 | 3 | 4 |
| ²√64 | 64 | 2 | 8 |
Enter your number (radicand) in the first box, enter your index (n) in the second box, and click Calculate. The tool returns your answer instantly, along with a simplified form where one exists.
The index is the small number that shows how many times a value must be multiplied by itself to reach the radicand. For square roots the index is 2 (usually left unwritten), for cube roots it's 3, and for any other root it's written explicitly.
It depends on whether your index is even or odd. Odd indexes (3, 5, 7...) give a real negative answer. Even indexes (2, 4, 6...) have no real answer for a negative radicand.
It's the standard radical symbol (√) with a small number placed in the notch to indicate the index — for example, ⁴√ for a 4th root or ⁵√ for a 5th root.
Most scientific calculators have a dedicated ˣ√y button, usually accessed through a shift or second-function key. If yours doesn't, you can raise your number to the power of (1 ÷ n) using the regular exponent button instead.
Yes — since square roots and cube roots are just the nth root with n = 2 and n = 3, this calculator works for those too. For dedicated tools, see our Racine Carrée Calculator and Racine Cubique Calculator.
Yes — no sign-up, no downloads, no usage limits.
An nth root generalizes one simple question — what number, multiplied by itself n times, gives you this value — to work for any index, not just squares and cubes. Once you understand the index/radicand relationship and the even-vs-odd rule for negative numbers, you can work with roots of any size. Use the calculator above for a fast, exact answer, and come back here anytime you want a refresher on how it all works.