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.
You already know square roots (index 2) and cube roots (index 3). But what happens when you need the 4th root, the 5th root, or any higher degree? That is exactly what this nth root calculator is built for.
An nth root generalizes roots to any number of multiplications. The nth root of a number asks: what value, multiplied by itself n times, produces this number?
The small number in the upper left of the radical sign. Dictates how many identical factors multiply together. If omitted, it is universally assumed to be 2 (square root).
The mathematical operator symbol (√). The horizontal upper bar, known as the vinculum, groups the radicand underneath to specify root extraction.
The number underneath the vinculum. When index n is even, the radicand must be non-negative (x ≥ 0) to yield a real-valued answer.
The non-negative real answer. For ⁴√81, the principal root is 3 because 3⁴ = 81. While (-3)⁴ also equals 81, radical notation strictly designates the principal root.
Example: The 4th root of 16 is 2, because 2 × 2 × 2 × 2 = 16. Roots of any degree share the same mathematical foundation.
The nth root is formally defined through the inverse of exponentiation:
If y = ⁿ√x, then yⁿ = x. Raising the root y to the power of n returns the radicand x.
√64 = 64^(1/2) = 8
∛125 = 125^(1/3) = 5
⁴√625 = 625^(1/4) = 5
⁵√1024 = 1024^(1/5) = 4
Whether a root accepts negative numbers depends strictly on whether the index is odd or even:
Select an index degree and radicand sign to see the mathematical outcome:
When the index is even and the radicand is positive, two real roots exist (±). The radical notation designates the positive principal root. Verified because 3⁴ = 81.
An even number of negative factors always multiplies to a positive number, while an odd number of negative factors preserves the negative sign.
Nth roots can be simplified by factoring out perfect nth powers:
Nth roots can be simplified by factoring out perfect nth powers:
The nth root is formally defined through the inverse of exponentiation:
Enter the positive whole-number index into the first box (e.g., 4 for a fourth root).
Enter the radicand into the second box.
Click Calculate to get your exact decimal and simplified radical form immediately.
Computer processors compute arbitrary roots iteratively using the recurrence relation:
Checking integer powers? Test for perfect square prime factor exponents →
The nth root uses the radical symbol (√) with a small integer n placed in the notch to indicate the index:
When no index is shown (√x), an index of 2 is implied. In fractional exponent form:
ⁿ√x = x^(1/n). Both notations represent the identical mathematical operation.
Key perfect powers and their exact integer roots:
| Root Degree (n) | Radicand (x) | Mathematical Form | Principal Root | Action |
|---|---|---|---|---|
| n = 2 | 16 | √16 = 16^(1/2) | 4 | |
| n = 2 | 49 | √49 = 49^(1/2) | 7 | |
| n = 2 | 144 | √144 = 144^(1/2) | 12 | |
| n = 3 | 27 | ∛27 = 27^(1/3) | 3 | |
| n = 3 | 64 | ∛64 = 64^(1/3) | 4 | |
| n = 3 | 125 | ∛125 = 125^(1/3) | 5 | |
| n = 3 | 1000 | ∛1000 = 1000^(1/3) | 10 | |
| n = 4 | 16 | ⁴√16 = 16^(1/4) | 2 | |
| n = 4 | 81 | ⁴√81 = 81^(1/4) | 3 | |
| n = 4 | 256 | ⁴√256 = 256^(1/4) | 4 | |
| n = 4 | 625 | ⁴√625 = 625^(1/4) | 5 | |
| n = 4 | 1296 | ⁴√1296 = 1296^(1/4) | 6 | |
| n = 5 | 32 | ⁵√32 = 32^(1/5) | 2 | |
| n = 5 | 243 | ⁵√243 = 243^(1/5) | 3 | |
| n = 5 | 1024 | ⁵√1024 = 1024^(1/5) | 4 | |
| n = 5 | 3125 | ⁵√3125 = 3125^(1/5) | 5 | |
| n = 6 | 64 | ⁶√64 = 64^(1/6) | 2 | |
| n = 6 | 729 | ⁶√729 = 729^(1/6) | 3 | |
| n = 6 | 4096 | ⁶√4096 = 4096^(1/6) | 4 | |
| n = 7 | 128 | ⁷√128 = 128^(1/7) | 2 | |
| n = 7 | 2187 | ⁷√2187 = 2187^(1/7) | 3 | |
| n = 8 | 256 | ⁸√256 = 256^(1/8) | 2 | |
| n = 8 | 6561 | ⁸√6561 = 6561^(1/8) | 3 | |
| n = 10 | 1024 | ¹⁰√1024 = 1024^(1/10) | 2 | |
| n = 10 | 59049 | ¹⁰√59049 = 59049^(1/10) | 3 |
Focusing specifically on degree 2 or 3? Use our dedicated step-by-step square root calculator or racine cubique calculator with negative number handling.
Nth roots are vital across advanced science, finance, and engineering:
CAGR uses the nth root: CAGR = ⁿ√(End/Start) − 1. A 5-year investment uses a 5th root; a 10-year investment uses a 10th root.
The geometric mean of n values is computed as ⁿ√(x₁·x₂·...·xₙ), ideal for growth rates and normalized indices.
Aerodynamics and materials science employ higher-order roots whenever equations feature nth-power dependencies.
Asymmetric cryptography algorithms (like RSA) rely on the computational difficulty of modular nth roots.
A square root asks what number multiplied by itself twice produces x (index 2). A cube root asks what number multiplied by itself three times produces x (index 3). Unlike square roots, cube roots of negative numbers have real solutions.
The index is the degree of the root (the number 'n' in ⁿ√x). An index of 2 is a square root, 3 is a cube root, 4 is a fourth root, and so on. It indicates how many times the root must multiply by itself to equal the radicand.
If the root degree n is odd (e.g., 3, 5, 7), a real negative solution exists. If the degree n is even (e.g., 2, 4, 6), there is no real solution because any real number raised to an even power is non-negative.
Yes, our calculators (Square Root, Cube Root, and Nth Root) are 100% free with unlimited calculations, no registration, and no download required.
An nth root answers what number multiplied by itself n times yields x. Odd indices accept negative numbers in real numbers; even indices require positive values. Use our calculator above for instant calculations.
