How this calculator works
The fifth-power function increases through both negative and positive values, so every real input has exactly one real fifth root. Take the fifth root of the magnitude and then restore its sign. For whole numbers with magnitude at most one billion, prime exponents in complete groups of five move outside the radical. Decimals and larger integers keep an unsimplified radical label. The signed-power table makes the difference between odd and even roots visible.
Formula
⁵√a = sign(a) × |a|^(1/5). If a = c⁵d, then ⁵√a = c × ⁵√d. The real equation x⁵=a has exactly one solution.
Worked example
-96 = (-2)⁵ × 3, so ⁵√(-96) = -2 × ⁵√3 ≈ -2.491461879. Since 3⁵=243, the fifth root of 243 is exactly 3. A fractional example is ⁵√(-1/32)=-1/2.
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Frequently asked questions
Why is a negative fifth root real?
Multiplying five negative factors leaves a negative product. Unlike an even power, a fifth power preserves the sign.
Why do some programming expressions return an error for negatives?
A generic fractional-power operation may use a different domain convention. Taking the positive magnitude first and restoring the sign selects the real odd root.
When is the radical reduced exactly?
For integers with magnitude up to one billion. The displayed decimal is approximate; the extracted-factor radical remains an exact mathematical expression.