√ Nth Root Calculator
By ToolNimba Math Team · Updated 2026-06-19
Enter a number and a root degree to compute the nth root.
The nth root of a number x is the value that, multiplied by itself n times, gives back x. This calculator finds any root you need: enter the number and the root degree (2 for a square root, 3 for a cube root, 4 for a fourth root, and so on) and it returns the result together with the steps. It also handles negative numbers correctly, showing a real answer for odd roots and flagging even roots of negatives as not real.
What is the Nth Root Calculator?
The nth root of x, written as the radical n-th-root of x, asks a simple question: what number raised to the power n equals x? For example the cube root of 8 is 2 because 2 x 2 x 2 = 8, and the fourth root of 81 is 3 because 3^4 = 81. Roots are the inverse operation of powers, so taking the nth root undoes raising to the nth power. The most reliable way to compute any root on a calculator is to use the exponent form: the nth root of x equals x^(1/n). That single identity turns every root into a power, which is exactly how this tool evaluates it.
The degree n controls how many equal factors you are splitting x into. A square root (n = 2) splits x into two equal factors, a cube root (n = 3) into three, and a tenth root (n = 10) into ten. As n grows the root of a number larger than 1 moves closer to 1, because you are spreading the same value across more factors. For a number between 0 and 1 the opposite happens: higher roots move toward 1 from below. Understanding this helps you sanity-check a result, the cube root of 1000 is 10, not 100, because 10 x 10 x 10 already reaches 1000.
Negative numbers need a little care. An odd root of a negative number is a real value, because an odd number of negative factors stays negative: the cube root of -27 is -3 since (-3) x (-3) x (-3) = -27. An even root of a negative number, however, has no real value, because any real number multiplied by itself an even number of times is positive or zero, so nothing squared (or raised to the fourth, sixth, and so on) can land on a negative. In that case the answer is a complex number, which this real-number calculator reports as not real rather than computing.
When to use it
- Finding the cube root or fourth root of a number for algebra, geometry or physics homework.
- Reversing a power: if a volume is x cubic units, the cube root gives the side length of the equivalent cube.
- Working out compound growth rates, the nth root of a ratio gives the constant per-period multiplier.
- Checking a value by hand when a basic calculator only offers a square root key and not an arbitrary root.
How to use the Nth Root Calculator
- Enter the number x you want to take the root of.
- Enter the root degree n (2 for square root, 3 for cube root, 4 for fourth root, and so on).
- Read off the nth root result shown in large type.
- Follow the step-by-step working, and use the built-in check that raises the answer back to the power n.
Formula & method
Worked examples
Find the cube root of 64 (x = 64, n = 3).
- Write it as a power: 64^(1/3)
- 1 / n = 1 / 3 = 0.333333
- 64^0.333333 = 4
- Check: 4 x 4 x 4 = 64
Result: The cube root of 64 is 4.
Find the 5th root of -32 (x = -32, n = 5).
- n = 5 is odd, so a real root of a negative number exists.
- Work on the magnitude: 32^(1/5) = 2
- Apply the sign because x is negative: result = -2
- Check: (-2)^5 = -32
Result: The 5th root of -32 is -2.
Find the 4th root of -16 (x = -16, n = 4).
- n = 4 is even.
- No real number raised to an even power can be negative.
- So there is no real 4th root of -16.
Result: The 4th root of -16 is not a real number.
Common nth roots of selected numbers
| Number x | Square root (n=2) | Cube root (n=3) | 4th root (n=4) |
|---|---|---|---|
| 16 | 4 | 2.5198 | 2 |
| 64 | 8 | 4 | 2.8284 |
| 81 | 9 | 4.3267 | 3 |
| 256 | 16 | 6.3496 | 4 |
| 1000 | 31.6228 | 10 | 5.6234 |
Whether a real nth root exists by sign of x and type of n
| Number x | n odd (3, 5, 7...) | n even (2, 4, 6...) |
|---|---|---|
| Positive | One real root | Real (principal, positive) |
| Zero | Root is 0 | Root is 0 |
| Negative | One real root (negative) | Not real |
Common mistakes to avoid
- Assuming every root of a negative number is undefined. Odd roots of negative numbers are perfectly real: the cube root of -27 is -3. Only even roots (square, fourth, sixth) of a negative number have no real value. Check whether n is odd or even before deciding.
- Confusing the nth root with dividing by n. The 4th root of 16 is 2, not 4. Taking a root is not the same as division. The root asks which number, used as n equal factors, multiplies up to x.
- Using a root degree of 0. The expression x^(1/n) needs n to be non-zero, since 1 divided by 0 is undefined. A root degree of 0 has no meaning, use 1 or higher.
- Expecting only one square root. Every positive number has two square roots, one positive and one negative, since both 3^2 and (-3)^2 equal 9. Calculators return the principal (positive) root by convention, so remember the negative root also exists.
Glossary
- Nth root
- The number that, raised to the power n, gives x. Written as the radical n-th-root of x, or as x^(1/n).
- Radicand
- The number under the radical sign, the value x whose root you are taking.
- Index (degree)
- The small number n on the radical that says which root to take: 2 for square, 3 for cube, and so on.
- Principal root
- The single root a calculator returns by convention, the non-negative one for an even root of a positive number.
- Radical
- The root symbol and the expression beneath it, representing a root operation.
Frequently asked questions
What is the nth root of a number?
The nth root of a number x is the value that, when raised to the power n, equals x. For example the cube root (n = 3) of 8 is 2 because 2 x 2 x 2 = 8. You can compute any nth root as x^(1/n).
How do I calculate a cube root?
A cube root is the nth root with n = 3. Enter your number, set the root degree to 3, and the calculator returns x^(1/3). For example the cube root of 125 is 5, since 5 x 5 x 5 = 125.
Can you take the root of a negative number?
It depends on the degree. An odd root (3, 5, 7...) of a negative number is real: the cube root of -27 is -3. An even root (2, 4, 6...) of a negative number is not a real number, it is complex, so this calculator reports it as not real.
What is the difference between a root and a power?
They are inverse operations. Raising to the power n multiplies a number by itself n times, while taking the nth root finds the value that would produce x when raised to that power. Because of this, the nth root of x equals x^(1/n).
Why does my answer have a long string of decimals?
Most roots are irrational, meaning they cannot be written as an exact fraction or terminating decimal. The square root of 2 and the cube root of 100 are examples. The calculator shows a rounded value accurate to several decimal places.
Does a positive number have more than one nth root?
In the real numbers, an even root of a positive number has two values, one positive and one negative (for example both 2 and -2 are fourth roots of 16). Calculators show the principal, non-negative root. An odd root has just one real value.