Nth Root Calculator — Advanced Options

Enter Values

Value (x)

Allows negative x when n is odd.

Degree (n)

Positive integer only.

Decimals

Method

Tolerance

Max Iterations

Show iterations log

History

Download CSV

Timestamp	x	n	Method	Tol	Iter	Dec	Result

History is kept in your session. CSV uses session data; PDF uses the on‑page table.

Formula Used

The real n‑th root of a value x is a real number y such that yⁿ = x. For x ≥ 0, the principal root is y = x^1/n. For x < 0 and odd n, y = −|x|^1/n. For x < 0 and even n, no real root exists.

We also implement Newton’s method for solving f(y) = yⁿ − x = 0:

y_k+1 = ((n−1)·y_k + x / y_kⁿ⁻¹) / n, with a tolerance stop criterion |Δy| ≤ tol · max(1, |y|).

How to Use

Enter the value x and a positive integer degree n.
Choose decimals, method, tolerance, and maximum iterations.
Optionally enable the iterations log to see convergence.
Press Calculate to get the root.
Every calculation is added to your on‑page history table.
Use Download CSV or Download PDF to export.

Example Data

x	n	Root	Note
27	3	3.000000000	Principal real root
16	4	2.000000000	Principal real root
10	3	2.154434690	Principal real root
-125	3	-5.000000000	Principal real root
0	5	0.000000000	Zero root
1000000	6	10.000000000	Principal real root
2	12	1.059463094	Principal real root

Perfect Powers Quick Reference (≤ 1000)

Squares

Numbers x with an exact square root y where x = y².

y	x = y^2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961

Cubes

Numbers x with an exact cube root y where x = y³.

y	x = y^3
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000

Fourth Powers

Numbers x with an exact fourth root y where x = y⁴.

y	x = y^4
1	1
2	16
3	81
4	256
5	625

Fifth Powers

Numbers x with an exact fifth root y where x = y⁵.

y	x = y^5
1	1
2	32
3	243

Use these lists to quickly spot inputs that yield exact integer roots.

Convergence Snapshot (Newton’s Method)

Iteration counts to reach tolerance illustrate expected performance on diverse inputs.

x	n	Tol	Max Iter	Iterations	Result	Status
27	3	1.0E-12	200	1	3	Converged
2	12	1.0E-12	200	1	1.059463094	Converged
1.0E-8	2	1.0E-12	200	1	0.0001	Converged
100000000	4	1.0E-12	200	1	100	Converged
-125	3	1.0E-12	200	1	-5	Converged
10	3	1.0E-12	200	1	2.15443469	Converged

Iteration counts depend on x, n, initial guess, and tolerance.

FAQs

There is no real solution because any even power of a real number is non‑negative. Consider complex numbers if required by your application.

Direct power uses exponentiation, while Newton iteratively solves yⁿ = x. Small numerical differences arise due to rounding, tolerance, and iteration count.

For most uses, 1e‑12 is a good start. If inputs are very large or very small, you may relax or tighten the tolerance based on stability.

This tool focuses on real roots. For complex results, use a math library supporting complex arithmetic or specialized software.

Extreme values can underflow or overflow floating‑point ranges. Consider scaling x, lowering n, or using higher precision arithmetic environments.

Displayed decimals affect presentation only. Internal computation uses full precision, then the result is rounded to your chosen number of decimals.

The initial guess is the principal root estimate. Each step applies Newton’s update until the change is within the chosen tolerance or the limit is reached.

y	x = y^2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961

y	x = y^2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961

Enter Values

History

Formula Used

How to Use

Example Data

Perfect Powers Quick Reference (≤ 1000)

Squares

Cubes

Fourth Powers

Fifth Powers

Convergence Snapshot (Newton’s Method)

FAQs

What happens with negative x and even n?

Why do results differ across methods?

What tolerance should I choose?

Can I compute complex roots?

Why is my result zero or infinity?

Does rounding affect accuracy?

How are iteration steps chosen?

Related Calculators

y	x = y^2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961