Fourth Root Calculator (⁴√x)
Calculate the fourth root (also known as quartic root) of any number with precision. This calculator handles both positive and negative numbers, identifies perfect fourth roots, and provides detailed step-by-step solutions.
Perfect Fourth Roots:
• 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
What is a Fourth Root?
The fourth root of a number is a value that, when raised to the fourth power, equals the original number. For example, 2 is the fourth root of 16 because 2⁴ = 16. Fourth roots are also known as quartic roots or biquadratic roots.
If ⁴√x = n, then n⁴ = x
⁴√16 = 2 because 2⁴ = 16
⁴√81 = 3 because 3⁴ = 81
Example Calculations
Perfect Fourth Roots
Small Numbers
- ⁴√1 = 1 (since 1⁴ = 1)
- ⁴√16 = 2 (since 2⁴ = 16)
- ⁴√81 = 3 (since 3⁴ = 81)
- ⁴√256 = 4 (since 4⁴ = 256)
- ⁴√625 = 5 (since 5⁴ = 625)
Larger Numbers
- ⁴√1,296 = 6 (since 6⁴ = 1,296)
- ⁴√2,401 = 7 (since 7⁴ = 2,401)
- ⁴√4,096 = 8 (since 8⁴ = 4,096)
- ⁴√6,561 = 9 (since 9⁴ = 6,561)
- ⁴√10,000 = 10 (since 10⁴ = 10,000)
Real-World Applications
Engineering
- Waveform analysis
- Signal processing
- Mechanical resonance
- Electrical circuit analysis
Mathematics
- Quartic equations
- Complex number theory
- Geometric calculations
- Polynomial factorization
Non-Perfect Fourth Roots
Irrational Results
⁴√50 ≈ 2.659240142516914
Verify:
(2.659240142516914)⁴ = 50
Properties of Fourth Roots
- Every positive real number has exactly one positive real fourth root
- Negative numbers have no real fourth roots (only complex roots)
- Perfect fourth powers have rational fourth roots
- Most fourth roots result in irrational numbers
- The fourth root function is strictly increasing for positive numbers
- Fourth roots are equivalent to taking a square root twice
Special Cases and Tips
- The fourth root of 0 is 0
- Negative numbers have no real fourth roots
- ⁴√(x⁴) = |x| for any real number x
- ⁴√(a·b) = ⁴√a · ⁴√b for positive numbers
- ⁴√(1/x) = 1/⁴√x for positive numbers