Calculate any nth root of any number. Supports square, cube, and higher roots. Shows radical simplification, fractional exponents, verification, and comparison tables.
The nth root of a number x, written ⁿ√x or x^(1/n), is the value r such that rⁿ = x. This is the general form that encompasses square roots (n=2), cube roots (n=3), fourth roots (n=4), and beyond. Understanding nth roots is foundational to algebra, calculus, and applied mathematics.
Every positive real number has a unique positive nth root for any positive integer n. For odd n, negative numbers also have a real nth root. The relationship between radicals and fractional exponents — ⁿ√x = x^(1/n) — is one of the most important identities in algebra, enabling simplification of radical expressions and solution of equations involving roots.
This general root calculator lets you compute any root of any real number with adjustable precision. It also simplifies radicals to their simplest form, generates tables of perfect nth powers, and offers a visual comparison of different root indices applied to the same number.
Understanding and computing nth roots is essential for students, engineers, and scientists. In algebra, roots are needed to solve polynomial equations. In geometry, they relate volumes and areas to edge lengths in higher dimensions. In finance, nth roots compute annualized returns over n-year periods.
This calculator handles all of these use cases in one tool — from simple square roots to exotic high-index roots — with radical simplification, verification, and visual comparison built in.
ⁿ√x = x^(1/n) Verification: (ⁿ√x)ⁿ = x Radical simplification: ⁿ√(aⁿ · b) = a · ⁿ√b Odd n: ⁿ√(−x) = −ⁿ√x
Result: 3
The cube root (3rd root) of 27 is 3 because 3³ = 27. In fractional exponent form, 27^(1/3) = 3.
Every positive real number has exactly one positive real nth root for each positive integer n. This is called the principal root. The existence of nth roots follows from the Intermediate Value Theorem: since f(r) = rⁿ is continuous and ranges from 0 to infinity, it must pass through every positive value. For odd n, the function also covers negative values, giving real roots for negative numbers.
Simplifying ⁿ√x means factoring out perfect nth powers. The process involves prime factorization: write x = p₁^a₁ · p₂^a₂ · … · pₖ^aₖ, then for each prime, extract ⌊aᵢ/n⌋ copies outside the radical and leave aᵢ mod n copies inside. For example, ∛(360) = ∛(8·45) = 2∛45 = 2∛(9·5) — but since ∛9 is not an integer, we get 2∛45.
Computers typically compute nth roots using the identity ⁿ√x = e^(ln x / n), leveraging fast logarithm and exponential algorithms. For manual computation, Newton's method with f(r) = rⁿ − x gives the iteration rₖ₊₁ = ((n−1)rₖ + x/rₖⁿ⁻¹)/n, which converges quadratically. In data science, nth roots appear in geometric means, normalization of high-dimensional data, and power transforms used in feature engineering.
The nth root of x is the number r such that rⁿ = x. It generalizes square roots (n=2) and cube roots (n=3) to any positive integer index.
ⁿ√x is identical to x^(1/n). This equivalence lets you use exponent rules to manipulate radicals.
Not in the real numbers. Even roots of negative numbers only exist in the complex number system. Odd roots of negative numbers are always real.
Simplifying means extracting perfect nth-power factors. For example, ∛(54) = ∛(27·2) = 3∛2.
Yes — the first root of any number is the number itself, since x¹ = x. Use this as a practical reminder before finalizing the result.
ⁿ√x = e^(ln(x)/n). Logarithms convert root calculations to division, which is how calculators compute them internally.
A number that equals an integer raised to the nth power. For n=3: 1, 8, 27, 64, 125, … are perfect cubes.