Calculate the nth root of any number instantly — enter a value and a root degree to get the result and a step-by-step solution.
What Is an nth Root?
The nth root of a number x is the value b that satisfies b^n = x.
In other words, the nth root reverses exponentiation: if 4^3 = 64, then the cube root of 64 is 4.
The most common examples:
| Root | Notation | Example |
|---|
| Square root | √x | √81 = 9, because 9² = 81 |
| Cube root | ∛x | ∛27 = 3, because 3³ = 27 |
| Fourth root | ⁴√x | ⁴√256 = 4, because 4⁴ = 256 |
The nth root of x is computed using the exponent rule:
ⁿ√x = x^(1/n)
Examples:
- √64 = 64^(1/2) = 8
- ∛125 = 125^(1/3) = 5
- ⁴√16 = 16^(1/4) = 2
- ⁵√32 = 32^(1/5) = 2
Special Cases
n = 1
The first root of any number is the number itself: ¹√x = x.
n = 2 — Square Root
The square root is the most common root. Every positive number has two square roots (positive and negative), but by convention the principal root (positive) is returned.
n = 3 — Cube Root
The cube root is defined for all real numbers, including negative ones:
- ∛8 = 2 (because 2³ = 8)
- ∛(−8) = −2 (because (−2)³ = −8)
Negative Numbers
- Odd n (1, 3, 5, …): the root of a negative number is a negative real number.
- Even n (2, 4, 6, …): the root of a negative number is a complex number — not supported by this calculator.
How to Calculate nth Root by Hand
For perfect powers, prime factorization works:
- Factor the number into prime factors.
- Group the factors into sets of n.
- Take one representative from each complete group; multiply them.
Example: ⁴√1296
- 1296 = 2⁴ × 3⁴
- One group of four 2s → 2; one group of four 3s → 3
- ⁴√1296 = 2 × 3 = 6 ✓
For non-perfect powers, use the formula x^(1/n) or a calculator.
Sources
- Wikipedia. Nth root.
- Weisstein, Eric W. Root. MathWorld — A Wolfram Web Resource.
Frequently Asked Questions
What is an nth root?
The nth root of a number x is the value b such that b^n = x. For example, the cube root of 27 is 3 because 3^3 = 27. The square root (n = 2) is the most familiar case: √9 = 3 because 3^2 = 9.
What is the formula for the nth root?
The nth root of x equals x raised to the power of 1/n: ⁿ√x = x^(1/n). This follows directly from the exponent rule (x^a)^b = x^(a×b): raising x^(1/n) to the power n gives x^(n/n) = x.
Can you take the nth root of a negative number?
It depends on n. If n is an odd integer (1, 3, 5, …), the nth root of a negative number is a negative real number — for example, ∛(−8) = −2 because (−2)^3 = −8. If n is even (2, 4, 6, …), the root of a negative number is a complex (non-real) number and this calculator shows an error.
What is the difference between a square root and a cube root?
A square root (n = 2) finds the number that, multiplied by itself, gives x: √25 = 5 because 5^2 = 25. A cube root (n = 3) finds the number that, used in a triple product, gives x: ∛64 = 4 because 4^3 = 64. Cube roots are defined for negative numbers; square roots are not (in real numbers).
What does it mean if the result is irrational?
Many roots are irrational numbers — they cannot be expressed as a simple fraction, and their decimal expansion never ends or repeats. For example, √2 ≈ 1.4142135… and ∛3 ≈ 1.4422495…. This calculator displays up to 10 significant digits.