Enter a number and a root index to get the nth root: the real root when it exists, the exact value for perfect powers, the simplified radical form, and for negative numbers the principal complex root. All n complex roots are listed and drawn on the complex plane.
Last updated:
2 = square root, 3 = cube root; decimals like 2.5 also work
Enter your values to see the result.
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How to use the Nth Root Calculator
Enter the number x (the radicand), positive or negative.
Enter the root index n: 2 for square root, 3 for cube root, and so on.
Choose the precision if you need more or fewer digits.
Read the real root, exact or simplified form, and the check.
See all complex roots in the table and on the complex-plane diagram.
Nth root formulas
Definition: ⁿ√x = r ⇔ rⁿ = x, and ⁿ√x = x^(1/n).
Using logarithms: ⁿ√x = e^(ln x / n) = 10^(log₁₀ x / n).
Product and quotient: ⁿ√(ab) = ⁿ√a · ⁿ√b and ⁿ√(a/b) = ⁿ√a / ⁿ√b (for non-negative a, b).
All complex roots: for x = |x|·e^(iθ), the roots are |x|^(1/n)·e^(i(θ + 2πk)/n) for k = 0, 1, …, n − 1.
Perfect powers reference
Base
Square
Cube
4th power
5th power
6th power
2
4
8
16
32
64
3
9
27
81
243
729
4
16
64
256
1,024
4,096
5
25
125
625
3,125
15,625
6
36
216
1,296
7,776
46,656
10
100
1,000
10,000
100,000
1,000,000
Read it backwards: the 5th root of 3,125 is 5, and the 6th root of 46,656 is 6.
Frequently asked questions
What is an nth root?
The nth root of x is a number r such that rⁿ = x. It is written ⁿ√x or x^(1/n). For example the 5th root of 243 is 3 because 3⁵ = 243, and the 4th root of 16 is 2 because 2⁴ = 16.
Can you take the nth root of a negative number?
If n is odd, yes: the cube root of −64 is −4 because (−4)³ = −64. If n is even there is no real root, because any real number raised to an even power is non-negative; the roots are complex, such as the 4th roots of −16, which are √2 ± √2·i and −√2 ± √2·i.
How many nth roots does a number have?
Every non-zero number has exactly n distinct complex nth roots. They lie on a circle of radius |x|^(1/n) in the complex plane, spaced 360°/n apart. A positive number has two real roots when n is even (±r) and one when n is odd.
How do I simplify an nth root?
Factor the number into primes and take out every group of n equal factors. For the 4th root of 80: 80 = 2⁴ × 5, so ⁴√80 = 2·⁴√5 ≈ 2.9907.
How do I calculate an nth root without a root key?
Use logarithms or powers: ⁿ√x = x^(1/n) = e^(ln x ÷ n). On most calculators you can type x ^ (1 ÷ n). Newton's method r ← r − (rⁿ − x)/(n·rⁿ⁻¹) refines a guess quickly.
What is the principal root?
For a positive number it is the positive real root. For a negative number with an even index, the principal root is the complex root with the smallest positive angle: |x|^(1/n)·(cos(π/n) + i·sin(π/n)).
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.