Fourth Root Calculator (4th Sqrt)
The fourth root of a number is the value that, multiplied by itself four times, gives the number back. It is also the square root of the square root. Enter any number and pick a root degree (4 is preselected) to get the decimal value, the simplified radical form (for example, the fourth root of 48 is 2∜3), and a verification that the answer raised to the power returns your input.
Quick Answer
The fourth root of a number is the value that, raised to the 4th power, gives back the number. This tool computes it as n to the power 1/4 and reports the decimal value, the simplified radical form, a square-root-of-square-root cross-check, and a verification line. For example, the fourth root of 48 is 2.6321480259, which simplifies to 2∜3, and 2.6321480259 raised to the 4th power equals 48. Enter your numbers above for the full breakdown.
Key takeaways
- The fourth root is the square root taken twice: ⁴√x equals √(√x).
- Simplification means pulling out perfect fourth powers: 48 equals 16 × 3, so ⁴√48 equals 2∜3.
- Even roots of negative numbers have no real value; odd roots of negatives do.
- Always check the answer by raising it back to the power. This tool prints that check for you.
- Quick Answer
- How the fourth root is actually computed
- Pulling perfect fourth powers out of the radical
- Common fourth roots worth recognizing
- Why negative numbers split by odd and even degrees
- Any root degree, not just four
- Where a fourth power hides in real work
- Factoring practice: three more numbers
- Fourth roots and fractional exponents
- What competing fourth-root tools leave out
- Frequently asked questions
- Add This Calculator to Your Website
- References
How the fourth root is actually computed
The tool raises your number to the power 1/4. That is the definition, not a shortcut: the fourth root of x is the number r such that r times r times r times r equals x, and fractional exponents write that as x to the power 1/4. It also cross-checks with the square root of the square root, which must agree digit for digit, because (x to the power 1/2) to the power 1/2 equals x to the power 1/4. When both routes land on the same decimal, you know the arithmetic is sound. One more rule matters: the radical sign means the principal root, the single non-negative answer. The equation r to the power 4 equals 16 has two real solutions, 2 and negative 2, but ⁴√16 means 2 only. Some competing tools print both signs as the answer, which is wrong for the principal root and confusing in homework.
Pulling perfect fourth powers out of the radical
Simplification is factoring with a purpose: find the largest perfect fourth power hiding inside the number, pull its root outside, and leave the rest under the radical. The LibreTexts treatment of roots and radicals works the same grouping idea for every index. Here is the whole method on the worked example.
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If no fourth-power factor exists, as with 7, the radical is already in simplest form and the tool says so plainly: ⁴√7. If the number is itself a perfect fourth power, the radical collapses to a whole number with nothing left inside. Our multiplying square roots calculator digs into how square roots combine and simplify, using the same factor-and-pull-out idea at index 2.
Common fourth roots worth recognizing
Perfect fourth powers collapse to whole numbers, and everything between them is an irrational decimal. That gap is exactly when the simplified radical form earns its keep: the decimal alone hides the structure, while the radical shows it.
| Number | Simplified form | Decimal |
|---|---|---|
| 1 | 1 | 1 |
| 16 | 2 | 2 |
| 32 | 2∜2 | 2.37841423 |
| 48 | 2∜3 | 2.6321480259 |
| 81 | 3 | 3 |
| 96 | 2∜6 | 3.1301691601 |
| 256 | 4 | 4 |
| 625 | 5 | 5 |
| 10000 | 10 | 10 |
Read the middle rows as factoring practice: 32 equals 16 × 2, and 96 equals 16 × 6. Pull out the fourth-power factor and the radical that remains is already simplest.
Why negative numbers split by odd and even degrees
No real number multiplied by itself four times gives a negative result, so even roots of negatives have no real value, and the tool refuses them and says why. Odd degrees are different: the cube root of negative 8 is genuinely negative 2, and the tool computes it. A purely imaginary answer would also be wrong: the four fourth roots of a negative are complex numbers with both real and imaginary parts, so refusing is the honest choice for a real-number tool. For a tool built around that odd-degree behavior, try our cube root calculator.
Any root degree, not just four
Change the degree to compute a square, cube, fifth, or any higher root the same way: decimal, exact-whole check, and verification. Fractional degrees work too, computed as v to the power 1/n: the 2.5th root of 100 is 6.3095734448, and raising it back to the 2.5 power returns 100. Note one honest difference: the simplified radical form is a fourth-root feature, so for other degrees the tool reports the exact-whole check instead (for example, the square root of 16 is exactly 4, while 7 has no perfect square root). Going the other direction, our exponential and logarithmic equations calculator solves the equations these roots come from.
Where a fourth power hides in real work
Anything that scales with the fourth power hides a fourth root in the reverse question. Thermal radiation is the cleanest example. The Stefan-Boltzmann law says the power a hot body radiates scales with the fourth power of its absolute temperature. Turn the question around, finding the temperature from the radiated power, and you need a fourth root: a body radiating 16 times the power per unit area of a 300 K body sits at 300 × ⁴√16, which is 600 K. Ratios make it clean: 81 times the radiated power means exactly 3 times the temperature. The same shape shows up whenever a quantity depends on a fourth power and you need to work backwards from an observed value to the cause, which is why the tool keeps the verification line visible: in applied work, a wrong root is worse than no root.
Factoring practice: three more numbers
The method never changes: pull out the largest perfect fourth power, then simplify what remains. First, ⁴√162. Since 162 equals 81 × 2, the answer is 3∜2, about 3.567621345. Second, ⁴√112. Since 112 equals 16 × 7, the answer is 2∜7, about 3.2531531234. Third, ⁴√1296, which is exactly 6, since 6 to the power 4 equals 1296. Perfect fourth powers up to 10 to the power 4 are worth recognizing on sight: 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, collapsing to 1 through 10. When a number is not on the list, factor it and let the tool check your hand work.
Fourth roots and fractional exponents
⁴√x and x to the power 1/4 are the same operation written two ways, and the exponent form explains why the root behaves the way it does. Raising the root back to the fourth power returns x because (x to the power 1/4) to the power 4 equals x to the power 4/4, which equals x, and that is exactly what the tool’s verification line checks. The exponent form also shows the root-of-a-root shortcut: x to the power 1/4 equals (x to the power 1/2) to the power 1/2, so ⁴√x equals √(√x), the square root taken twice. And it explains the sign rule in one line: a negative base to a fractional power with an even denominator has no real value, while an odd denominator keeps the sign. Khan Academy’s nth roots lesson builds the same connection from the radical side.
What competing fourth-root tools leave out
Most fourth-root tools print a decimal and stop. That decimal is hard to check, easy to mistype, and useless when the exact form is what the problem asks for. Others cover only the fourth root and force a second search for square or cube roots, or they blur the principal root by printing both signs. This page does all four things in one place: the simplified radical form, the square-root-of-square-root cross-check, the verification line, and an any-degree mode with honest negative handling. If another tool gives you a different decimal for the same input, raise its answer back to the fourth power and see which one returns your number.
Frequently asked questions
About 2.6321480259, exactly 2∜3, since 48 equals 16 times 3 and the fourth root of 16 is 2.
Factor out perfect fourth powers: 48 equals 16 times 3, so the fourth root of 48 equals 2 times the fourth root of 3.
3, since 3 times 3 times 3 times 3 equals 81.
There is none in the real numbers. Odd roots of negatives are real, even roots are not.
The fourth root of x equals the square root of the square root of x, because taking a square root twice is the same as raising to the power one quarter.
1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, and 10000, which collapse to the whole numbers 1 through 10.
Yes. Change the root degree to 2, 3, 5, or any value 2 or higher, including fractional degrees such as 2.5.
Raising the answer back to the fourth power must return your input, so the verification line proves the decimal is right instead of asking you to trust it.
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References
- Wolfram MathWorld: Radical (Accessed Oct 25, 2026).
- Paul's Online Notes: Radicals (Accessed Oct 25, 2026).
- Purplemath: Simplifying Radicals (Accessed Oct 25, 2026).
- Khan Academy: Nth Roots (Accessed Oct 25, 2026).
- OpenStax: Simplify Square Roots (Accessed Oct 25, 2026).
- LibreTexts: Roots and Radicals (Accessed Oct 25, 2026).
- Math is Fun: nth Root (Accessed Oct 25, 2026).
- LibreTexts: Roots and Radicals (Engineering) (Accessed Oct 25, 2026).
- SaveMyExams: Stefan-Boltzmann Law (Accessed Oct 25, 2026).