Fifth Root Calculator — ⁵√x
Find the fifth root (⁵√x) of any real number — the value that, raised to the fifth power, gives x. Because 5 is an odd exponent, the fifth root of a negative number is also negative, which makes this calculator more general than even or fourth roots.
⁵√x — the number whose fifth power equals x
- 1
Apply x^(1÷5)
32^(1÷5) = 2 - 2
Verify: (⁵√x)⁵
(2)⁵ = 32Should equal the original input; tiny rounding differences in the last digit are normal.
How does this calculator work?
The fifth root of x is x^(1/5) — the number that, multiplied by itself five times, equals x. Enter any real number (negative numbers are supported since 5 is odd). Result is rounded to 6 decimal places; the verification row confirms (⁵√x)⁵ ≈ x.
Formula
How this is calculated
The fifth root of a number x is the value r such that r⁵ = x. Mathematically this is computed as x^(1/5), or equivalently exp(ln(x)/5) for positive x. Because 5 is an odd integer, the function is defined for all real numbers — negative inputs simply give a negative fifth root, mirroring the result for positive inputs. The calculator raises |x| to the power 1/5 and then restores the sign.
The verification row ((⁵√x)⁵) should equal x exactly, subject to floating-point rounding — this is a useful sanity check. For large or very small inputs, small rounding errors in the last decimal place are normal and do not indicate a mistake.
The curve y = ⁵√x is a monotonically increasing odd function, symmetric about the origin. It grows more slowly than a square root for x > 1 and more steeply near zero. The interactive chart marks the computed point on the curve so you can visually confirm where your input sits.
Frequently asked questions
Estimate by trial and improvement: guess r, compute r⁵, adjust up or down, and repeat. For example, ⁵√32 — try r = 2: 2⁵ = 32. For non-integer inputs, successive approximations converge quickly. Newton's method (r ← r − (r⁵ − x)/(5r⁴)) converges in a few iterations.
Yes. Unlike square or fourth roots, the fifth root of a negative number is a real (negative) number because an odd power preserves sign. For example, ⁵√(−32) = −2, since (−2)⁵ = −32. This is a key difference from even-order roots, which have no real result for negative inputs.
Both are odd roots of real numbers. The fifth root extracts a smaller value than the cube root for the same input — for instance ⁵√100 ≈ 2.51 while ³√100 ≈ 4.64. The fifth root grows more slowly because it represents dividing the exponent by 5 instead of 3. Both are continuous odd functions defined on all real numbers.
Also known as
TG we-Calculate Editorial Team. (2026). Fifth Root Calculator — ⁵√x [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/fifth-root-calculator
TG we-Calculate Editorial Team. "Fifth Root Calculator — ⁵√x." TG we-Calculate. 2026. https://we-calculate.com/calculator/fifth-root-calculator.
TG we-Calculate Editorial Team, "Fifth Root Calculator — ⁵√x," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/fifth-root-calculator
@misc{wecalculate_fifth_root_calculator, title = {Fifth Root Calculator — ⁵√x}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/fifth-root-calculator}}, year = {2026}, note = {TG we-Calculate} }
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