Fourth Root Calculator
Calculate the fourth root of any number with step-by-step explanations and prime factorization
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Enter any real number to find its root
Fourth Root Result
Common Fourth Roots
Root Calculation Tips
The fourth root of a number x is the number that, when multiplied by itself 4 times, gives x
⁴√a = a^(1/4)
Fourth root is always positive for positive numbers
Fourth roots of negative numbers are complex (not real)
Prime factorization helps identify perfect fourth powers
Understanding Fourth Roots
What is a Fourth Root?
The fourth root of a number is a value that, when multiplied by itself four times, equals the original number. It's the inverse operation of raising a number to the 4th power.
Mathematical Definition
If a = b⁴, then b = ⁴√a
Examples
- ⁴√16 = 2, because 2⁴ = 16
- ⁴√81 = 3, because 3⁴ = 81
- ⁴√256 = 4, because 4⁴ = 256
Calculation Methods
1. Prime Factorization
Break the number into prime factors and group them in sets of 4.
20736 = 2⁸ × 3⁴ = (2²)⁴ × 3⁴
Therefore: ⁴√20736 = 2² × 3 = 12
2. Exponential Form
Use the relationship ⁿ√a = a^(1/n)
⁴√512 = 512^(1/4) ≈ 4.757
3. Estimation
For large numbers, estimate by finding nearby perfect fourth powers.
Important Notes
- • Fourth roots of negative numbers are complex (not real numbers)
- • Not all numbers have rational fourth roots
- • Perfect fourth powers have integer fourth roots
- • The fourth root function is continuous and strictly increasing for positive numbers