Nth Root Calculator
Calculate real nth roots with a positive whole-number index. Supports zero and negative radicands for odd n, with worked examples and four-decimal results.
Formula: ⁿ√x = x^(1/n); for negative x, n must be odd
Use a positive whole number for n. Negative inputs are supported only for odd n; results are rounded to four decimal places.
The Nth Root Calculator finds a real nth root from two inputs: a finite number x and a positive whole-number index n. Positive numbers and zero are supported for every valid n. A negative number is supported only when n is odd, because an even root of a negative number is not a real number. Results are displayed to four decimal places.
With the default inputs 32 and n = 5, the result is 2.0000 because 2⁵ = 32. The calculator evaluates the entered numbers; it does not generate a symbolic expression or a complex-number answer.
How to Use the Nth Root Calculator
- Enter the number whose root you want in the Number field.
- Enter n as a positive whole number in the Root (n) field.
- Select Calculate to view the real nth root.
Changing either input clears the previous result until you calculate again. Decimal radicands are supported. Use a positive whole number for n: it must be 1 or greater and cannot contain a fractional part. When n = 1, the first root is the number itself, so −8 with n = 1 returns −8.0000.
Nth Root Formula and Worked Examples
ⁿ√x = y when yⁿ = x
For x ≥ 0: ⁿ√x = x^(1/n)
For x < 0 and odd n: ⁿ√x = −((-x)^(1/n))OpenStax College Algebra 2e defines principal nth roots for positive-integer indices n ≥ 2 and relates radicals to rational exponents. This widget additionally accepts n = 1 as the identity case, where the result equals x.
For a negative example, −8 with n = 3 returns −2.0000 because (−2)³ = −8. Likewise, −32 with n = 5 returns −2.0000. Zero with n = 2 returns 0.0000 because every positive-integer root of zero is zero.
Input Rules, Real-Number Domain and Browser Limits
Blank, nonnumeric and non-finite entries are invalid. n = 0, a negative n and a fractional n are rejected. A negative x with an even n is rejected in the Number field because no real result exists. This calculator does not return complex roots.
Calculations use standard JavaScript floating-point numbers. MDN documents the largest finite JavaScript Number; magnitudes outside the finite range are not accepted. Results are rounded for display to four decimal places, so the displayed digits do not add precision to the inputs. If a calculation does not produce a finite real result, the widget clears the result and displays an accessible calculation error.
Use the Square Root Calculator for n = 2 or the Cube Root Calculator for n = 3. The Exponent Calculator performs the inverse operation by raising a base to a power.
Frequently asked questions
What does an nth root mean?
The nth root of x is a real value y such that y raised to the power n equals x. For nonnegative x, it can be written as ⁿ√x = x^(1/n).
Why must n be a positive whole number?
n is the root index. This calculator requires an integer of at least 1, so zero, negative and fractional values of n are rejected.
Can I find an nth root of a negative number?
Yes, when n is odd. For example, the cube root of −8 is −2. A negative number with an even n has no real nth root and is rejected.
What happens when n equals 1?
The first root is the identity case: the result equals the entered number. For example, the first root of −8 is −8.
What is the nth root of zero?
It is zero for every valid positive-integer n. For example, the square root of zero is 0.0000 in the four-decimal display.
How are results rounded, and what input limits apply?
Results are displayed to four decimal places using JavaScript floating-point arithmetic. Inputs must be finite JavaScript numbers; out-of-range values are rejected, and display rounding does not add precision.
Does this calculator return complex roots?
No. It returns real-number results only. A negative radicand with an even index has complex roots, so that input combination is rejected.