4th Root Calculator
Calculate the fourth root of any non-negative number instantly. Enter a value to get the exact fourth root, free online.
How to Use the Fourth Root Calculator
- Enter the number (x) you want to find the fourth root of.
- Click the “Calculate” button.
- View the fourth root instantly.
Features of the Fourth Root Calculator
- Instantly calculates the fourth root of any non-negative real number.
- Works with whole numbers and decimals.
- Clearly flags when no real fourth root exists, rather than showing a confusing error value.
- Free to use with no sign-up required.
Fourth Root Formula
The fourth root of a number x is the value that, when multiplied by itself four times, equals x.
Fourth Root = ⁴√x
Since 4 is an even number, fourth roots are only defined for non-negative numbers, because no real number multiplied by itself four times can produce a negative result.
Common Fourth Roots (Reference Table)
| Number (x) | Fourth Root (⁴√x) |
|---|---|
| 1 | 1 |
| 16 | 2 |
| 81 | 3 |
| 256 | 4 |
| 625 | 5 |
| 10000 | 10 |
| 0.5 | 0.840896 |
| 2 | 1.189207 |
| 10 | 1.778279 |
How to Calculate a Fourth Root by Hand
To find the fourth root of a perfect fourth power, identify which whole number multiplied by itself four times gives the original number. For numbers that aren’t perfect fourth powers, the fourth root is found using estimation, a calculator, or the exponent 1/4.
Suppose you want the fourth root of 81. Since 3 × 3 × 3 × 3 = 81, the fourth root of 81 is 3.
Thus, ⁴√81 = 3.
Solved Examples on Fourth Roots
1. Fourth Root of a Perfect Fourth Power
Example: Find the fourth root of 256.
Since 4 × 4 × 4 × 4 = 256, ⁴√256 = 4.
2. Fourth Root of a Non-Perfect Fourth Power
Example: Find the fourth root of 10.
10 is not a perfect fourth power, so ⁴√10 ≈ 1.778279.
3. Fourth Root of a Decimal
Example: Find the fourth root of 0.5.
⁴√0.5 ≈ 0.840896.
4. Fourth Root of a Negative Number
Example: Find the fourth root of −16.
No real number, when raised to the fourth power, produces a negative result, so ⁴√−16 has no real solution.
1. What is a fourth root?
A fourth root of a number x is a value that, when multiplied by itself four times, produces x. It is written as ⁴√x or x^(1/4).
2. Can negative numbers have fourth roots?
Not as real numbers. Since 4 is even, raising any real number (positive or negative) to the fourth power always produces a non-negative result, so a negative number can never be a perfect fourth power in the real number system.
3. How is a fourth root different from a fifth root?
A fourth root is an even root, so it’s only defined for non-negative numbers. A fifth root is an odd root, so it’s defined for all real numbers, including negatives, since a negative number raised to an odd power stays negative.
4. What is a perfect fourth power?
A perfect fourth power is a number that results from multiplying an integer by itself four times, such as 1, 16, 81, 256, and 625. Their fourth roots are always whole numbers.
5. Does a number have more than one fourth root?
Every positive real number has two real fourth roots, one positive and one negative, since both a number and its negative give the same positive result when raised to an even power. This calculator returns the principal (positive) fourth root.
6. Where are fourth roots used in practice?
Fourth roots appear in statistics when working with fourth-power relationships, in engineering for scaling and stress calculations, and in algebra when simplifying radical expressions involving even powers.