Case 1
- Number (radicand)
- 27
- Root degree (n)
- 3
3.000000
Open with these valuesNth Root Calculator
3.000000
Result: 3.000000The nth root of a number is the value that, raised to the power n, gives that number back. Taking a root is therefore the same as raising to the power 1/n: the cube root of 27 is 27^(1/3) = 3. Enter the number and the degree — 2 for a square root, 3 for a cube root.
Worked examples
Case 2
- Number (radicand)
- 2
- Root degree (n)
- 2
1.414214
Open with these valuesCase 3
- Number (radicand)
- 256
- Root degree (n)
- 8
2.000000
Open with these valuesHow it's calculated
ⁿ√x = x^(1/n)
- StepEnter the number you want the root of.
- StepEnter the degree: 2 for a square root, 3 for a cube root, and so on.
- ResultRead the value that, raised to that power, returns your number.
Reference table
| Number, degree | Root | Result |
|---|---|---|
| 0, 5 | Fifth root of zero | 0.000000 |
| 1, 7 | Any root of one | 1.000000 |
| 2, 2 | Square root of two, irrational | 1.414214 |
| 16, 4 | Fourth root | 2.000000 |
| 256, 8 | Eighth root | 2.000000 |
| 27, 3 | Cube root | 3.000000 |
| 16, 2 | Square root | 4.000000 |
| 100, 2 | Square root | 10.000000 |
Questions
How do I calculate the nth root of a number?
Raise the number to the power of one over the degree: ⁿ√x = x^(1/n). The cube root of 27 is 27^(1/3) = 3, because 3 × 3 × 3 = 27.
What is an nth root?
The nth root of a number x is the value that, raised to the power n, gives back x. It is the inverse of raising to a power: the square root undoes squaring, the cube root undoes cubing. A degree of 2 means a square root and 3 means a cube root.
Why is a root the same as a fractional exponent?
Because raising to the power 1/n is defined as taking the nth root, so that (x^(1/n))ⁿ = x^(n/n) = x. Writing roots as fractional exponents lets one power operation compute every root, which is exactly what this calculator does.
Can I take the root of a negative number?
Only for an odd degree, and this calculator does not cover that case: its input starts at zero. An even root of a negative number has no real value at all, and mixing both rules into one field would hide that. The cube root of −8 is −2, worked out by hand.
Why can the degree not be zero?
Because the nth root is defined as x^(1/n), and 1/0 is undefined — there is no zeroth root. The degree here starts at 1, which simply returns the number itself.
Sources and last check
Information, not professional advice.