Nth Root
Extract root of any degree.
What does Nth Root mean?
The nth root of a number x is a value that, when raised to the power n, gives x. For example, the square root (2nd root) of 144 is 12 because 12² = 144, and the cube root (3rd root) of 27 is 3 because 3³ = 27. Roots are the inverse of exponentiation and are fundamental in algebra, geometry, physics, and engineering — from computing distances with the Pythagorean theorem to normalizing data in statistics.
How to calculate Nth Root
The nth root of x is calculated as x^(1/n), or equivalently Math.pow(x, 1/n). For example, the 4th root of 81 = 81^(1/4) = 3, because 3⁴ = 81. For negative radicands, odd roots are real (e.g., ³√(−8) = −2), while even roots produce complex numbers and are undefined in real arithmetic.
FAQ
A square root (2nd root) finds a value that, when multiplied by itself, gives the original number (e.g., √9 = 3). A cube root (3rd root) finds a value that, when cubed, gives the original number (e.g., ³√27 = 3). The key difference is the degree: square roots use n = 2, cube roots use n = 3.
Yes, but only for odd roots. Odd roots of negative numbers are real and negative (e.g., ³√(−8) = −2, because (−2)³ = −8). Even roots of negative numbers (e.g., √(−4)) are not real numbers — they produce complex (imaginary) results.
Nth roots appear in many areas: the Pythagorean theorem uses square roots to find distances, cube roots help compute volumes, geometric means use nth roots to average ratios, and financial formulas use roots to annualize multi-year returns (e.g., CAGR). They are also used in physics, signal processing, and data normalization.
Taking the nth root is the inverse of raising to the nth power. If aⁿ = x, then ⁿ√x = a. In exponential notation, ⁿ√x = x^(1/n). Logarithms provide another inverse: if bⁿ = x, then log_b(x) = n. So roots and logarithms are both ways to "undo" exponentiation, but they solve for different unknowns.
These are verification values. Result² shows your root raised to the power of 2, and Result³ shows it raised to the power of 3. When computing a square root, Result² should equal your original radicand; for a cube root, Result³ should match. This helps you verify the calculation is correct.
Related calculators
- Logarithm— Inverse exponent function.
- Quadratic Formula— Solve second-degree equation.
- Factorial— Product of descending integers.