Real nth roots and principal even roots
The result is a real value whose nth power equals the input, subject to floating-point rounding. An even-degree radical denotes the principal nonnegative root, although an equation such as x² = 81 has two real solutions.
To raise a value to a specified power, use the Exponent Calculator .
The formula
Square and cube roots use dedicated numerical operations. Other degrees use a fractional power of the absolute input, restoring a negative sign for odd degrees. Even roots of negative numbers would require complex arithmetic and are rejected.
Worked example
The square root of 81 is 9. The cube root of −125 is −5, because (−5)³ = −125. The fourth root of 16 is 2, while the equation x⁴ = 16 also has the real solution −2.
How to use this calculator
- Enter the radicand, which is the number under the root sign.
- Choose degree 2 for a square root, 3 for a cube root, or another integer from 1 to 1,000.
- Calculate and check the stated root convention before using the result.
Principal roots and equation solutions
A radical is a function with a single selected value. For a positive radicand and an even degree, this calculator selects the nonnegative value. Finding every solution of an equation is a different task. Do not automatically attach a plus-or-minus sign to an odd root, because the opposite sign would produce the opposite radicand.
Negative radicands
An odd number of negative factors has a negative product, so an odd root of a negative number is real. An even number of identical negative factors has a positive product. This is why −125 has a real cube root but −81 does not have a real square root. The tool reports that boundary directly.
Using a root in another calculation
Retain more digits than the final measurement requires when transferring an approximate root into a formula. A displayed value such as 1.4142135624 is a rounded representation of √2. Root degree 1 returns the input itself. Zero has root zero for every supported positive integer degree.
Assumptions & limitations
What this calculation assumes
- Degree is a positive integer; results are real numbers.
What to keep in mind
- No complex roots, exact radical simplification, or list of all polynomial solutions.
Common questions
Can I calculate a negative root degree?
This tool uses positive degrees. A negative power belongs in the exponent calculator and introduces a reciprocal.
Why is the square root of 81 shown as 9 rather than ±9?
The radical √81 denotes the principal root. Both 9 and −9 solve x² = 81, but that equation asks a different question.