The fifth root reverses raising a number to the fifth power. It is an odd root, like the cube root, so it works on negative numbers as well as positive ones, and every real number has exactly one real fifth root. Fifth roots show up most often in finance, where they convert a five-year total into an average yearly rate. This calculator finds them exactly when possible and to ten decimal places otherwise.
How to use the fifth root calculator
- Enter the radicand: any whole number, decimal, fraction or negative number.
- Read the fifth root on the tape. Perfect fifth powers (32, 243, 100,000 …) are reported exactly; other numbers show the simplest radical form and a decimal.
- The steps factor the radicand into primes and pull out complete groups of five.
Fifth root formulas
The product and quotient rules work just as they do for square roots: ⁵√(ab) = ⁵√a · ⁵√b and ⁵√(a/b) = ⁵√a / ⁵√b.
Worked example: ⁵√96
Factor: 96 = 2⁵ × 3.
Group in fives: the five 2s form one complete group and come out as a single 2. The 3 stays inside.
Result: ⁵√96 = 2·⁵√3 ≈ 2 × 1.2457309396 = 2.4914618792.
Check: 2⁵ = 32 and 3⁵ = 243, and 96 is between them, so the root is between 2 and 3. ✓
A negative example: ⁵√−243 = −3, because (−3)⁵ = −243. And ⁵√100,000 = 10, since 10⁵ = 100,000.
Perfect fifth powers
| n | n⁵ | n | n⁵ |
|---|---|---|---|
| 1 | 1 | 6 | 7,776 |
| 2 | 32 | 7 | 16,807 |
| 3 | 243 | 8 | 32,768 |
| 4 | 1,024 | 9 | 59,049 |
| 5 | 3,125 | 10 | 100,000 |
A neat fact: n⁵ always ends in the same digit as n. So the fifth root of 16,807 must end in 7, and since 16,807 is between 10⁵ and 20⁵, the root is 7.
Fifth roots and five-year growth
The compound annual growth rate (CAGR) over n years is the n-th root of the total growth factor, minus 1. For five years:
An account grew from $10,000 to $16,105.10 in five years. The growth factor is 16,105.10 ÷ 10,000 = 1.61051.
⁵√1.61051 = 1.1, so the CAGR is 1.1 − 1 = 10% per year.
Simply dividing the 61.05% total gain by 5 would suggest 12.2% per year, which overstates the true compound rate.
The CAGR calculator does this for any number of years, and the nth root calculator takes roots of any index.
Estimating a fifth root by hand
- Bracket the number between perfect fifth powers. For 1,000: 3⁵ = 243 and 4⁵ = 1,024, so the root is just under 4.
- Refine with Newton’s method: g → (4g + x/g⁴) ÷ 5. Starting at g = 4 for x = 1,000 gives (16 + 1,000/256) ÷ 5 ≈ 3.981, already close to the true 3.98107.
Odd and even roots compared
| Odd roots (∛, ⁵√, …) | Even roots (√, ∜, …) | |
|---|---|---|
| Negative radicand | real, negative | not real |
| Real solutions of xⁿ = a > 0 | one | two (±) |
For even roots see the fourth root calculator; for odd ones, the cube root calculator.
Frequently asked questions
What is a fifth root?
The fifth root of x is the number that gives x when raised to the fifth power. ⁵√32 = 2 because 2 × 2 × 2 × 2 × 2 = 32. In exponent form it is x^(1/5).
Can a negative number have a fifth root?
Yes. Five is odd, so an odd number of negative factors gives a negative product: (−3)⁵ = −243, which means ⁵√−243 = −3. Every real number has exactly one real fifth root.
How is the fifth root used in finance?
It turns a five-year growth factor into an average annual rate. If a value grew from $10,000 to $16,105.10 in five years, the factor is 1.61051 and its fifth root is 1.1, so the compound annual growth rate was 10%.
How do I write a simplified fifth root?
Pull out every group of five equal prime factors. 96 = 2⁵ × 3, so ⁵√96 = 2·⁵√3. The small 5 is the index of the radical, not an exponent on the 2.