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How to use
The nth root reverses raising a number to the power n. Over the real numbers, a negative radicand has a real nth root only when n is an odd integer.
- Enter x.
- Enter integer n from 2 to 1000.
- Review the root and approximate power-back check.
How it is calculated
Example
For the fourth root of 81, find the non-negative number whose fourth power is 81: 3×3×3×3=81, so ⁴√81=3. The equation y⁴=81, however, has two real solutions, 3 and −3; the calculator returns the principal root only. For an odd index, ∛(−8)=−2 because (−2)³=−8. Roots need not be integers: √2≈1.41421356.
Important notes
The power-back check can differ slightly from x because browser floating-point arithmetic is finite-precision.
Worked examples and interpreting results
An nth root is a number whose nth power returns the original input. The cube root of 27 is 3, the fourth root of 16 is 2, and the cube root of −27 is −3.
| Case | Calculation | Result |
|---|---|---|
| Cube root | ³√27 | 3 |
| Principal fourth root | ⁴√16 | 2 |
| Odd root of a negative number | ³√(−27) | -3 |
How to check the result
An even root of a negative number is not real. The principal fourth root of 16 is 2, although x⁴=16 has two real solutions, ±2. The root order must be a positive integer.
Frequently asked questions
Does an even root of a negative number have a real value?
No, not in the real-number domain.
Are negative cube roots supported?
Yes, because 3 is odd.
Can n be decimal?
No. This tool uses an integer root index.
How can I check an approximate root?
Raise the result to the root index and compare it with the original number. Keep as many available digits as possible; rounding the root before raising it to a large power can amplify the difference.