Compute the fourth root of a number with step-by-step formula ⁴√x = x^(1/4) = √(√x).
Common fourth-root values
| x | ⁴√x |
|---|---|
| 0.0001 | 0.1 |
| 1 | 1 |
| 16 | 2 |
| 81 | 3 |
| 256 | 4 |
| 625 | 5 |
| 1,296 | 6 |
| 2,401 | 7 |
| 4,096 | 8 |
| 6,561 | 9 |
| 10,000 | 10 |
How it works
How the fourth root is computed
The fourth root of x is the value y such that y⁴ = x. It can be written as x^(1/4) or x^0.25. A quick way to compute it by hand is to take two consecutive square roots: ⁴√x = √(√x). For example, ⁴√10000 = √(√10000) = √100 = 10. The calculator uses JavaScript's Math.pow(x, 0.25) for positive inputs and -Math.pow(Math.abs(x), 0.25) would give a real cube-like mirror — but the fourth root of a negative is not real, so we report this and show the real fourth root of |x| as a reference.
Relation to the nth root
The fourth root is a specific case of the nth root: ⁿ√x with n = 4. It is also the square root of the square root. In index form: ⁴√x = x^(1/4). The fourth power undoes it: if y = ⁴√x, then y⁴ = x. This makes the fourth root the functional inverse of raising to the 4th power for non-negative inputs.
Frequently asked questions
What is a fourth root?
The fourth root of a number x is the value that, when multiplied by itself four times, equals x. For example, ⁴√81 = 3 because 3 × 3 × 3 × 3 = 81. In exponent form it is x^(1/4) — raising x to the one-quarter power.
Can the fourth root of a negative number be a real number?
No. Any real number raised to an even power (second, fourth, sixth, and so on) is non-negative, so no real number raised to the fourth power can equal a negative value. The fourth root of a negative number is a complex number. The calculator flags this and, for reference, also shows the real fourth root of the absolute value.
How is the fourth root different from the square root?
The square root (²√x) is the value y where y² = x; the fourth root (⁴√x) is the value y where y⁴ = x. The fourth root is the square root of the square root: ⁴√x = √(√x). So ⁴√16 = √(√16) = √4 = 2, while √16 = 4.
How can I compute a fourth root manually?
Three practical methods:
- Double square root. Take √x, then take √ of that result. Example: ⁴√625 → √625 = 25 → √25 = 5.
- Exponent form. Compute x^0.25 on any scientific calculator using the y^x or x^y key.
- Logarithms. log(⁴√x) = (1/4) × log(x). Take the log, divide by 4, then take the antilog (10^result).
What are some exact fourth roots worth knowing?
Useful reference values: ⁴√1 = 1, ⁴√16 = 2, ⁴√81 = 3, ⁴√256 = 4, ⁴√625 = 5, ⁴√1296 = 6, ⁴√2401 = 7, ⁴√4096 = 8, ⁴√6561 = 9, ⁴√10000 = 10. Any perfect fourth power n⁴ has an exact integer fourth root n.
Enter any number to get its fourth root (⁴√x) to up to 12 decimal places. The tool shows the step-by-step derivation ⁴√x = x^(1/4) = √(√x) and a table of perfect fourth powers for quick reference. Examples: ⁴√16 = 2 because 2 × 2 × 2 × 2 = 16; ⁴√81 = 3 because 3⁴ = 81; ⁴√10000 = 10. For negative inputs the calculator explains that a real fourth root does not exist (the value would be complex) and displays the real fourth root of the absolute value for reference. A copy-to-clipboard button, adjustable decimal precision, and thousands-separator options help when pasting results into spreadsheets, physics problems, or statistics work such as RMS calculations and variance fourth-moment analysis.