Fifth Root Calculator
Calculate the fifth root of any number with step-by-step explanations and prime factorization
Calculate Root
Enter any real number to find its root
Fifth Root Result
Common Fifth Roots
Root Calculation Tips
The fifth root of a number x is the number that, when multiplied by itself 5 times, gives x
⁵√a = a^(1/5)
Odd roots (3rd, 5th, 7th...) can handle negative numbers
Even roots (2nd, 4th, 6th...) of negative numbers are complex
Prime factorization helps identify perfect powers
Understanding Fifth Roots
What is a Fifth Root?
The fifth root of a number is a value that, when multiplied by itself five times, equals the original number. It's the inverse operation of raising a number to the 5th power.
Mathematical Definition
If a = b⁵, then b = ⁵√a
Examples
- ⁵√32 = 2, because 2⁵ = 32
- ⁵√243 = 3, because 3⁵ = 243
- ⁵√7776 = 6, because 6⁵ = 7776
Calculation Methods
1. Prime Factorization
Break the number into prime factors and group them in sets of 5.
7776 = 2⁵ × 3⁵ = (2×3)⁵
Therefore: ⁵√7776 = 2×3 = 6
2. Exponential Form
Use the relationship ⁿ√a = a^(1/n)
⁵√125 = 125^(1/5) ≈ 2.6265
3. Estimation
For large numbers, estimate by finding nearby perfect fifth powers.
Important Notes
- • Fifth roots of negative numbers are real (unlike square roots)
- • Not all numbers have rational fifth roots
- • Perfect fifth powers have integer fifth roots
- • The fifth root function is continuous and strictly increasing