Nth Root Calculator

Calculate the nth root of any number with step-by-step solution and simplified form.

Quick:

What Is the Nth Root?

The nth root of a number x is a value r such that r raised to the power n equals x. In mathematical notation, this is written ⁿ√x or x1/n. When n = 2 we get the familiar square root; when n = 3, the cube root; and so on for any positive integer n.

Every positive real number has exactly one positive nth root, called the principal nth root. For negative radicands, a real result exists only when n is odd — for example, ∛(−8) = −2 because (−2)³ = −8. When n is even and x is negative, the result is complex (outside the scope of this calculator).

The radical symbol (√) with an index is one of the oldest notations in algebra, dating back to 16th-century Europe. Today, fractional exponents are often preferred in higher mathematics: ⁿ√x = x1/n, which slots neatly into the laws of exponents.

How to Use This Calculator

  1. Enter the radicand — the number whose root you want to find.
  2. Enter the root index n — for square root use 2, cube root use 3, etc.
  3. Click Calculate to see the result instantly with a full step-by-step breakdown.
  4. Use the quick preset buttons (√, ∛, ∜, ⁵√) to jump to common roots.
  5. Try the example buttons (√144, ∛216, ⁵√1024) to see the calculator in action.

Nth Root Formula & Examples

ⁿ√x = x1/n

The nth root is equivalent to raising x to the power of 1/n.

ExpressionnxResultVerification
√14421441212² = 144
∛216321666³ = 216
∜8148133⁴ = 81
⁵√10245102444⁵ = 1024
√2221.41421…Irrational
∛(−27)3−27−3(−3)³ = −27

Common Nth Roots Reference Table

x√x (n=2)∛x (n=3)∜x (n=4)⁵√x (n=5)
11111
421.5871.4141.319
82.82821.6821.516
1642.52021.741
275.19632.2801.933
64842.8282.297
100104.6423.1622.512
100031.623105.6233.981

Real-World Applications of Nth Roots

Finance & Investing

Compound annual growth rate: CAGR = (FV/PV)1/n − 1, where n is the number of years. This uses the nth root to find the equivalent annual return.

Geometry

Finding the side length of an n-dimensional hypercube with known volume V: side = ⁿ√V. Cube side = ∛V is the familiar 3D case.

Statistics

The geometric mean of n numbers is the nth root of their product: G = ⁿ√(x₁·x₂·…·xₙ). Used in growth rates and index numbers.

Physics & Engineering

Scaling laws often involve fractional exponents. The frequency of a resonator scales as the square root of stiffness; power dissipation uses cube roots in fluid dynamics.

Frequently Asked Questions

The nth root of a number x is a value r such that rⁿ = x. Written ⁿ√x or x^(1/n). For example, the cube root (n=3) of 27 is 3, because 3³ = 27. For n=2 you get the square root; every higher n generalises the concept further.
Use the fractional exponent rule: ⁿ√x = x^(1/n). On a scientific calculator, enter x, press the yˣ or ^ button, then enter (1÷n). For example, the 5th root of 1024 = 1024^(0.2) = 4. You can verify by computing 4⁵ = 1024.
The square root is a special case of the nth root where n = 2. It finds r such that r² = x. The nth root generalises this to any index: it finds r such that rⁿ = x. Square root (n=2), cube root (n=3), fourth root (n=4), and so on are all nth roots.
Yes, but only for odd values of n. When n is odd (1, 3, 5, …), the nth root of a negative number is also negative. For example, ∛(−8) = −2, because (−2)³ = −8. When n is even (2, 4, 6, …), the nth root of a negative number is not a real number — it is an imaginary/complex number.
The nth root of 0 is 0 for any positive n, since 0^n = 0. The nth root of 1 is 1 for any positive n, since 1^n = 1. Both 0 and 1 are "fixed points" of the root operation — no matter what root index you use, you get the same value back.
The nth root is exactly equivalent to a fractional exponent: ⁿ√x = x^(1/n). More generally, ⁿ√(x^m) = x^(m/n). This means every root can be rewritten as a power, allowing the laws of exponents (product rule, quotient rule, chain rule) to apply directly to radicals.
Nth roots appear across many fields: finance (computing CAGR uses nth root), geometry (side of an n-dimensional box from its volume), statistics (geometric mean = nth root of product), signal processing (RMS calculations), and physics (scaling laws like how resonant frequency scales with mass).

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