How this instrument works
A fifth root undoes the operation of raising a number to the fifth power: ⁵√x is the one real number that, multiplied by itself five times, rebuilds x. Because five is odd, the underlying curve y = x⁵ climbs without ever turning back — its slope, 5x⁴, never dips below zero — so every real number, negative ones included, has exactly one real fifth root. That single fact splits fifth root from its even-numbered neighbor: a fourth root behaves like an ordinary square root and refuses negative inputs outright, because y = x⁴ folds back on itself and assigns the same output to both x and −x. Fifth root, like the cube root already on this site, keeps the sign attached to a unique answer instead.
The growth of this curve is slower than most people expect. Doubling the fifth root of 1, which is 1, requires multiplying the input all the way up to 32, since 2⁵ = 32. Push further and the pattern only sharpens: reaching a fifth root of 3 needs x = 243, and a fifth root of 10 needs x = 100,000. Five multiplications compound quickly on the way up, so undoing them with a fifth root compresses an enormous input into a comparatively small, well-behaved output — a useful thing to know before assuming two numbers are close just because their fifth roots look similar.
Extracting the fifth root of one number, as this sheet does, is nothing more than solving x⁵ = a for a single unknown, and that is straightforward. Solving a general fifth-degree equation with several mismatched terms is a different matter entirely: unlike the quadratic, cubic, and quartic, no formula built only from addition, multiplication, and radicals can solve every quintic, a fact Niels Henrik Abel proved in 1824, building on earlier work by Paolo Ruffini. The single radical ⁵√x sits on the easy side of that boundary — it is a complete equation solved for x, not a general polynomial with lower-power terms standing in the way.
- Type any real number into the x field — positive, negative, or zero all work identically, with no special handling required on your end.
- Read the result in the ⁵√x field: the one real number that, raised to the fifth power, reproduces x exactly.
- Try a negative entry such as −32 to see the sign carried through automatically instead of an error or a complex value.
- Adjust x and watch ⁵√x update instantly — useful for checking how slowly the output grows against a rapidly rising input.
Worked example — a five-stage amplifier
A five-stage amplifier applies the same gain factor at every stage, and the combined boost measured across all five stages comes to 32-fold. To find the per-stage gain, set x = 32: the sheet returns ⁵√x = 2, meaning each stage multiplies the signal by exactly 2. The check runs backward without rounding — 2 × 2 × 2 × 2 × 2 comes to 32 on the nose, so the answer is exact, not an approximation trimmed to a few decimal places.
Flip the sign to see the same machinery handle a case a fourth-root sheet cannot: enter x = −32 and it returns ⁵√x = −2, because (−2) multiplied by itself five times comes to −32. A fourth root would refuse that entry outright, since raising any real number to an even power can never land on a negative result — but a real fifth root exists for every point on the number line, negative ones included, which is exactly why the sign is kept separate from the magnitude inside the formula.
Questions
What is a fifth root?
The fifth root of x is the one real number that, multiplied by itself five times, gives back x. ⁵√32 = 2 because 2 × 2 × 2 × 2 × 2 = 32, and ⁵√(−32) = −2 because (−2)⁵ = −32. It is the direct inverse of raising a number to the fifth power, the same relationship division has to multiplication.
Why does a negative number have a real fifth root when a negative number has no real fourth root?
Raising to an even power erases sign — both 2⁴ and (−2)⁴ equal 16, so no single real value can undo it once the result is negative. Raising to an odd power keeps the sign: (−2)⁵ comes to −32, not 32, so ⁵√(−32) can trace it straight back to −2. Every real number, positive, negative, or zero, has exactly one real fifth root; only non-negative numbers have a real fourth root.
Why can't a spreadsheet just raise x to the power one fifth when x is negative?
General fractional powers are usually evaluated through logarithms, and logarithms of negative numbers are undefined on the reals, so x^(1/5) applied directly to a negative x often errors out or returns a complex value in spreadsheets and most programming languages. This sheet avoids that by extracting the root from the size of x and reattaching the original sign afterward, matching the true real-valued answer.
Is there a formula for solving a general fifth-degree equation like the quadratic formula?
No, and that is one of the more surprising results in algebra. The quadratic, cubic, and quartic all have general solving formulas built from radicals, but Niels Henrik Abel proved in 1824 that no such formula exists for every quintic. Extracting ⁵√x here is different: it solves the single-term equation x⁵ = a, which radicals handle easily; the impossibility only strikes equations with several mismatched powers of x.
How many fifth roots does a number actually have?
Five, once complex numbers are allowed — but only one of them is real, and that real value is what this sheet returns. The other four sit spaced evenly around the complex plane, each rotated 72 degrees from the real one, and only matter once a problem has moved past ordinary real-number arithmetic.
Does the fifth root grow slower than the cube root and square root for large numbers?
Yes, for any input above 1. A hundred thousand has a fifth root of exactly 10, a cube root near 46.4, and a square root of about 316 — the higher a root's degree, the more the input has to grow before its output moves noticeably. It also explains why doubling ⁵√1 up to ⁵√32 = 2 needs an input 32 times larger, far more than the 8-times jump a cube root needs for the same doubling.