Fifth Root Calculator

What a fifth root is

The fifth root of a number $x$ is the value that, multiplied by itself five times, produces $x$:

$\sqrt[5]{x} = y \quad \text{means} \quad y \times y \times y \times y \times y = x$

Since $2 \times 2 \times 2 \times 2 \times 2 = 32$, the fifth root of $32$ is $2$ — written $\sqrt[5]{32} = 2$. A fifth root undoes raising a number to the fifth power, the same way a cube root undoes cubing, just two powers higher.

Negative numbers have a real fifth root — unlike square or fourth roots

This is the key similarity with cube roots (and the key difference from square and fourth roots): a negative number does have a real fifth root. Raising to the fifth power is an odd power, so it preserves the sign of the input — a negative number to the fifth stays negative, with no sign ambiguity to resolve:

$\sqrt[5]{-32} = -2 \quad \text{because} \quad (-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32$

Compare this to the fourth root, an even index, where no real number raised to the fourth power gives a negative value, forcing an imaginary result. A fifth root never needs the imaginary unit $i$ — every real number, positive, negative, or zero, has exactly one real fifth root. This calculator computes negative inputs directly as real numbers.

Perfect fifth powers vs. simplified radicals

A perfect fifth power is a whole number that is some integer raised to the fifth power — $1, 32, 243, 1024, 3125, \ldots$ Its fifth root is an exact whole number:

Most numbers are not perfect fifth powers, so their fifth root is irrational — a never-ending, non-repeating decimal. Rather than truncate that decimal, the exact answer is written as a simplified radical: a whole-number coefficient times the smallest possible root. For example, $\sqrt[5]{64} = 2\sqrt[5]{2}$, because $64 = 32 \times 2$ and $\sqrt[5]{32} = 2$ can be pulled out from under the root, leaving $2$ — which has no fifth-power factors left to extract.

Simplifying a radical, step by step

To simplify $\sqrt[5]{64}$ by hand:

1. Factor into primes: $64 = 2^6$.
2. Group into fives: one group of five 2s ($2^5$) forms a complete group; the single leftover $2$ has no group to join.
3. Pull each group out as its fifth root: $2^5 \to 2$, which becomes the coefficient.
4. Leave the leftover factor under the root: the unpaired $2$ stays inside, giving $\sqrt[5]{64} = 2\sqrt[5]{2}$.

This calculator shows that same prime-factorization breakdown for whatever number you enter — including negative numbers, where the sign carries straight through to the coefficient (e.g. $\sqrt[5]{-64} = -2\sqrt[5]{2}$).

The inverse: raising to the fifth power

Taking a fifth root and raising to the fifth power undo each other: $(\sqrt[5]{x})^5 = x$ for any real $x$. If $\sqrt[5]{y} = x$, then $x^5 = y$ — so $\sqrt[5]{1024} = 4$ and $4^5 = 1024$ are the same fact, read in opposite directions. Use the Exponents calculator to compute $x^5$ directly.

Worked examples

Example 1 — perfect fifth power: $\sqrt[5]{243} = 3$, exactly, since $3^5 = 243$.

Example 2 — simplified radical: $\sqrt[5]{64} = 2\sqrt[5]{2} \approx 2.297397$, since $64 = 32 \times 2$.

Example 3 — negative input: $\sqrt[5]{-32} = -2$, a real result (not imaginary), since $(-2)^5 = -32$.

Example 4 — already simplified: $\sqrt[5]{2} \approx 1.148698$ stays as $\sqrt[5]{2}$ — 2 has no fifth-power factors to pull out.

Where fifth roots show up

Fifth roots appear whenever a quantity that scales by the fifth power needs to be reversed — some growth models and higher-dimensional geometry problems use fifth-power relationships, and a value multiplying five-fold every 5 periods implies a per-period rate of $\sqrt[5]{5}$. Raising to the fifth power — the exact inverse — is covered by the general Exponents calculator, the Cube Root calculator covers the other odd-index case (also real for negatives), and the Fourth Root calculator covers the neighboring even-index case where negative inputs become imaginary instead of real.