The fourth root asks which number, multiplied by itself four times, produces a given value. Because a fourth power is a square of a square, the fourth root is simply the square root taken twice, which makes it easy to estimate. This calculator gives the exact answer for perfect fourth powers, the simplest radical form otherwise, and a ten-place decimal in every case.
How to use the fourth root calculator
- Enter the number under the radical. Whole numbers, decimals and fractions are accepted.
- Read the fourth root on the tape, with the simplest radical form, both real roots (±) and the two whole numbers the answer lies between.
- The steps show the prime factorization and how complete groups of four equal factors leave the radical.
Fourth root formulas
| Rule | Formula | Example |
|---|---|---|
| Product | ∜(ab) = ∜a · ∜b | ∜162 = ∜81 · ∜2 = 3∜2 |
| Quotient | ∜(a/b) = ∜a / ∜b | ∜(1/16) = 1/2 |
| Nested square roots | ∜x = √(√x) | ∜256 = √16 = 4 |
| Decimals | shift the point in groups of four | ∜0.0081 = 0.3 |
Worked example: ∜162
Factor: 162 = 2 × 3⁴.
Group in fours: the four 3s form a complete group and come out as a single 3. The 2 has no partners and stays inside.
Result: ∜162 = 3∜2 ≈ 3 × 1.1892071150 = 3.567621345.
Check by square roots: √162 ≈ 12.7279, and √12.7279 ≈ 3.5676. ✓
Perfect fourth powers come out exactly: ∜10,000 = 10 and ∜0.0081 = 0.3, since 0.3 × 0.3 × 0.3 × 0.3 = 0.0081.
Perfect fourth powers
| n | n⁴ | n | n⁴ |
|---|---|---|---|
| 1 | 1 | 6 | 1,296 |
| 2 | 16 | 7 | 2,401 |
| 3 | 81 | 8 | 4,096 |
| 4 | 256 | 9 | 6,561 |
| 5 | 625 | 10 | 10,000 |
Fourth powers grow quickly, so a number between two of these has a fourth root between the corresponding values of n. For example 2,000 lies between 1,296 and 2,401, so ∜2,000 is between 6 and 7 (it is about 6.687).
Why even roots need care
Fourth powers can never be negative, so two rules follow:
- No real fourth roots of negatives. ∜−16 is not a real number. Its principal complex value, √2 + √2·i, is what the calculator reports.
- Two real solutions to x⁴ = a. For a > 0, both +∜a and −∜a work. When solving equations, write x = ±∜a; when evaluating the radical, give only the positive root.
Where fourth roots appear
- Thermal radiation. The Stefan–Boltzmann law says radiated power grows with the fourth power of absolute temperature, so a star’s temperature is proportional to the fourth root of its power output per unit area. Doubling the output raises temperature by only ∜2 ≈ 1.19.
- Average growth over four periods. If an investment grows by a total factor of 1.4641 over four years, the average yearly factor is ∜1.4641 = 1.1, which is 10% per year.
- Radar range. A radar’s maximum detection range grows with the fourth root of transmitter power, so 16 times the power only doubles the range.
For other indexes, try the fifth root calculator or the nth root calculator, and for radicals with a coefficient, the simplify radicals calculator.
Frequently asked questions
What is a fourth root?
The fourth root of x is the non-negative number that, raised to the fourth power, equals x. ∜81 = 3 because 3⁴ = 3 × 3 × 3 × 3 = 81. It can also be written x^(1/4).
Is the fourth root the same as taking the square root twice?
Yes, for non-negative numbers. ∜x = √(√x), because (x^(1/2))^(1/2) = x^(1/4). For 10,000, √10,000 = 100 and √100 = 10, so ∜10,000 = 10.
What is the fourth root of a negative number?
There is no real fourth root of a negative number, since any real number to the fourth power is non-negative. The calculator reports the principal complex root instead; for −16 that is √2 + √2·i ≈ 1.414 + 1.414i.
Why does the calculator show ± for fourth roots?
The equation x⁴ = 81 has two real solutions, 3 and −3, because (−3)⁴ = 81 as well. The radical ∜81 by convention means only the positive one, 3.