Nth Root Calculator
Calculate any root (2nd, 3rd, 4th, 5th, etc.) with step-by-step solutions, complex roots using De Moivre's theorem, and perfect power detection.
x^(1/n) = y where y^n = xResult
4th Root
3
Enter Values
The number under the radical
2 = square root, 3 = cube root, etc.
Number of decimal places
Display all n complex roots using De Moivre's theorem
4sqrt(81) = 3
Both real roots: 3 and -3
Step-by-Step Solution
Given radicand: 81
Root index (n): 4
Formula: n-th root of x = x^(1/n)
4-th root of 81 = 81^(1/4)
= 3
81 is a perfect 4th power!
3^4 = 81 = 81
Note: Even roots have two real solutions: +3 and -3
Verification: (3)^4 = 81.0000000000 = 81
Result
4th Root
3
?How Do You Calculate an Nth Root?
The nth root of x is written as x^(1/n) or the nth radical of x. It finds the number y such that y^n = x. For example, the 4th root of 81 is 3 because 3^4 = 81. Even roots of negative numbers are complex (imaginary), while odd roots of negatives are real. Every number has n distinct nth roots in the complex plane, found using De Moivre's theorem.
What is an Nth Root?
The nth root of a number x, written as x^(1/n) or with the radical symbol with index n, is a value y such that y^n = x. The principal nth root is the positive real root for positive x. For negative x, odd roots are real and negative, while even roots are complex. Using De Moivre's theorem, all n complex nth roots can be found, equally distributed around a circle in the complex plane.
Key Facts About Nth Roots
- nth root of x: find y where y^n = x
- 4th root of 16 = 2 because 2^4 = 16
- 5th root of 32 = 2 because 2^5 = 32
- Even roots of negatives are complex
- Odd roots of negatives are real: 5th root of -32 = -2
- Every number has exactly n nth roots (including complex)
- x^(1/n) x x^(1/n) x ... (n times) = x
- De Moivre: complex roots are evenly spaced on a circle
Frequently Asked Questions
The nth root of x is a number y such that y^n = x. For example, the 4th root of 16 is 2 because 2^4 = 16. It's written as x^(1/n) or with a radical symbol with index n.
Even roots (2nd, 4th, 6th, etc.) of negative numbers are not real; they are complex/imaginary. For example, the 4th root of -16 requires complex numbers. However, odd roots of negatives are real.
Every non-zero number has exactly n different nth roots in the complex plane. For positive numbers with even n, there are also 2 real roots (positive and negative). Odd roots have 1 real root.
De Moivre's theorem provides a formula for finding all n complex nth roots. For a number with magnitude r and angle theta, the nth roots have magnitude r^(1/n) and angles (theta + 2*pi*k)/n for k = 0 to n-1.
A perfect nth power is an integer that equals another integer raised to the nth power. Examples: 16 is a perfect 4th power (2^4), 32 is a perfect 5th power (2^5), 64 is both a perfect square (8^2) and a perfect cube (4^3).
Last updated: 2025-01-15
Result
4th Root
3