About the Nth Root Calculator
This nth root calculator finds the number that, raised to the power n, gives your number. Enter the radicand x and the index n — 2 for a square root, 3 for a cube root, 4 for a fourth root, or any other value, including decimals — and it returns the real root, checks it by raising it back to the nth power, and tells you whether x is a perfect nth power.
For whole numbers it also simplifies the radical, pulling perfect nth powers outside: ⁴√48 becomes 2·⁴√3 and ∛250 becomes 5·∛2, which is the exact form algebra courses ask for. For whole-number indexes up to 12 it lists all n complex roots (the nth roots of unity scaled by the real root), which matters in algebra, trigonometry and engineering.
Negative numbers are handled correctly: odd roots of negatives are real (∛−27 = −3), while even roots of negatives have no real value, so the calculator gives the principal complex root instead. It is useful for students, for geometric mean and growth-rate calculations, and for anyone who needs a root beyond the √ key.
With the default inputs, the real nth root is 2.6321480259. Change any value above to recalculate instantly.
How to use the nth root calculator
- 1Enter the number x you want the root of.
- 2Enter the index n, e.g. 3 for a cube root or 5 for a fifth root.
- 3Read the real root and the check value.
- 4Use the simplified radical form for exact answers.
- 5See the table for all n complex roots if you need them.
Formula and method
The nth root of x is the number r with rⁿ = x, written ⁿ√x or x^(1/n). For x ≥ 0 there is one non-negative real root. For x < 0 a real root exists only when n is an odd integer, and it is −ⁿ√|x|; when n is even the real root does not exist. To simplify a radical, factor x into primes and move each group of n identical factors outside the root: 48 = 2⁴ · 3, so ⁴√48 = 2·⁴√3.
In the complex numbers every non-zero x has exactly n nth roots. Writing x = |x|·e^(iθ), with θ = 0 for positive and π for negative x, the roots have magnitude ⁿ√|x| and angles (θ + 2πk)/n — they sit evenly spaced around a circle. The principal root is the one with k = 0.
- x
- Radicand — the number under the root
- n
- Index of the root (2 = square, 3 = cube…)
- θ
- Angle of x in the complex plane (0 or π for real x)
- k
- Which of the n complex roots (0 … n − 1)
Worked examples
Fourth root of 48
48 = 2⁴ × 3, so ⁴√48 = 2·⁴√3 ≈ 2 × 1.31607 = 2.63215. Raising 2.63215 to the 4th power gives back 48.
Cube root of a negative number: ∛−27
Because (−3)³ = −27 and 3 is odd, the real cube root of −27 is −3. Its other two cube roots are the complex numbers 1.5 ± 2.598i.
Fifth root of 100,000
10⁵ = 100,000, so the fifth root is exactly 10 — a perfect fifth power.
Even root of a negative: ⁴√−16
No real number to the 4th power is negative. In the complex plane the four fourth roots of −16 have magnitude 2 at angles 45°, 135°, 225° and 315°, i.e. ±√2 ± √2·i.
Frequently asked questions
What is an nth root?+
The nth root of x is the number that gives x when multiplied by itself n times. The cube root of 64 is 4 because 4 × 4 × 4 = 64. It can also be written as a fractional exponent: ⁿ√x = x^(1/n).
Can you take the root of a negative number?+
An odd root of a negative number is a real negative number (∛−8 = −2). An even root of a negative number is not real — ⁴√−16 has only complex values such as √2 + √2·i.
How do you simplify an nth root?+
Factor the radicand into primes, group identical factors into sets of n, and move one factor from each group outside the root. For ∛250: 250 = 5³ × 2, so ∛250 = 5∛2.
How do I calculate an nth root on a regular calculator?+
Use the power key with a fractional exponent: x^(1/n). For the fifth root of 200, type 200 ^ (1 ÷ 5) ≈ 2.8854. Wrap 1/n in brackets so the calculator divides before raising the power.
How many nth roots does a number have?+
Every non-zero number has exactly n complex nth roots, spaced evenly around a circle in the complex plane. Positive numbers have one positive real root (plus a negative one when n is even); the rest are complex.
What is the difference between a square root and a 4th root?+
A square root undoes squaring (n = 2) and a fourth root undoes raising to the fourth power (n = 4). The fourth root is the square root of the square root: ⁴√81 = √9 = 3.