Fifth root calculator
The fifth root of 32 is 2, and the fifth root of −32 is −2, because an odd root keeps the sign. Enter any finite number and get its one real fifth root, with perfect fifth powers returned as exact whole numbers. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 32
- Fifth root
- 2
The fifth root of a number x is the unique real number y such that y to the fifth power equals x. Because five is odd, every real radicand has exactly one real fifth root: the fifth root of thirty-two is two, the fifth root of negative thirty-two is negative two, and the fifth root of zero is zero. People search for a dedicated fifth root calculator when they need that one function, not a generic nth-root panel, a square-root tool that rejects negatives, or a cube-root tool with a different index. This capability accepts one finite number, computes the real fifth root, and returns that value together with the echoed radicand and a short formula so a client can show the identity next to the answer. Failed validation is not billed.
What the real fifth root means and how it is computed
The fifth root of a real number x is the unique real y satisfying y to the fifth power equals x. Five is an odd positive integer, so the real map x to x to the one-fifth is defined on the entire real line and is strictly increasing: it sends negatives to negatives, zero to zero, and positives to positives.
That is the algebraic fact students meet when they first invert a fifth-power monomial, and it is why a fifth-root tool must not copy the JavaScript Math.pow shortcut that returns NaN for a negative base with a fractional exponent. Square root is different: the principal square root is real only for a non-negative radicand.
Cube root is the closer sibling, because three is also odd, but cube root and fifth root are distinct search intents and distinct functions — the fifth root of thirty-two is two, while the cube root of thirty-two is not an integer. This calculator isolates the fifth-root case.
Internally it takes the sign of the radicand, computes a positive fifth root of the absolute value by Newton iteration on y to the fifth minus the absolute value equals zero (the update is four y to the fifth plus the absolute value, all divided by five y to the fourth, with a positive power initial guess), then reapplies the sign.
Perfect fifth powers of small integers snap to the exact integer so that the fifth root of thirty-two is two and the fifth root of negative two hundred forty-three is negative three rather than a noisy neighbour of those values.
What you can do with it
Fifth-power inverses in scripts and notebooks
Invert a fifth-power scaling law in automation without importing a full CAS library or hitting Math.pow NaN on a negative base.
FAQ
Does the fifth root work for negative numbers?
Yes. Five is odd, so every real x has exactly one real fifth root. The fifth root of -32 is -2, and the fifth root of -243 is -3. Square root is the function that rejects negatives.
What happens if the input is not a finite number?
Missing, empty, non-numeric, NaN, and infinite radicands are rejected and the failure is not billed.
What is the fifth root of 32, 0, and 1?
The fifth root of 32 is 2 because 2^5 = 32. The fifth root of 0 is 0. The fifth root of 1 is 1. Perfect fifth powers of small integers snap to the exact integer.
How is the fifth root different from the cube root?
Both have an odd index, so both accept negative numbers, but they are different functions. The fifth root of 32 is exactly 2, while the cube root of 32 is not a whole number.
How precise is the fifth root result?
Results are rounded to twelve decimal places, and perfect fifth powers such as 243 snap to the exact whole number, 3, instead of a noisy value right next to it.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/fifth-root \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"value":32}'const res = await fetch("https://api.kit.forhosting.com/math/fifth-root", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"value": 32
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/fifth-root",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"value": 32
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/fifth-root", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"value":32}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"value":32}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/fifth-root", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"value": 32
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.fifth_root",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
A missing, empty, non-numeric, NaN, or infinite input fails with invalid_input instead of leaking a silent NaN that a homework grader might treat as success. A successful API call costs $0.002.
What the real fifth root means and how it is computed
There is no random seed, no wall-clock timestamp, and no network call; identical inputs always produce identical JSON.
Input contract, response shape, and error behaviour
POST a JSON body with one required finite number named value: the radicand x whose real fifth root you want. Aliases n, number, x, num, input, and radicand are accepted so homework scripts and science notebooks can keep a familiar field name without remapping. The field may arrive as a JSON number or as a numeric string in ordinary decimal or scientific notation; missing fields, empty strings, booleans, objects, arrays, NaN, and Infinity fail with invalid_input naming the field. On success the body echoes value with minus-zero collapsed to zero, returns fifth_root as the real fifth root rounded to twelve decimal places, and includes formula as a short identity that records y to the fifth equals x and the odd-index sign rule. Classic checks: value thirty-two yields fifth_root two because two to the fifth is thirty-two; value negative thirty-two yields fifth_root negative two because five is odd; value zero yields zero; value one yields one; value negative one yields negative one; value two hundred forty-three yields three. Pair the response with your own plotting code if you need a sketch of y equals x to the one-fifth; this endpoint computes the real fifth root only and does not emit graphs, complex fifth roots of unity, or a generic nth-root for a caller-chosen index. Validation failures do not consume prepaid balance. This is not a square-root calculator, not a cube-root calculator, and not a modular nth-root tool over a prime field — those are different landings with different contracts. Keep the request small and pin the rounded fifth_root field in tests so a later change to rounding cannot pass unnoticed.
Where a dedicated fifth root calculator earns its keep
Classrooms and tutoring platforms use a fifth root calculator so learners can check the real fifth root for arbitrary finite radicands, including the negative cases that some graphing calculators in the wrong mode silently refuse. Engineering and science notebooks invert fifth-power scaling laws — volume-like quantities raised to one-fifth, some allometric relations, and certain polynomial models — without pulling a full computer-algebra system for a one-argument root. Failed validation does not consume balance. Exam generators and LMS auto-graders need a deterministic oracle that rejects a non-finite radicand as invalid_input instead of writing NaN into a markbook.
Integration use cases
Classroom and homework checks
Let students submit a finite radicand and verify the real fifth root, including textbook values such as fifth_root(32)=2 and fifth_root(-32)=-2.
Exam and LMS auto-grading
Mark fifth-root items with a deterministic oracle that rejects missing or non-finite radicands as invalid_input rather than writing NaN.
Technical questions
How do I compute the fifth root of a number?
Send a finite number value (aliases n, number, x, num, input, radicand). The API returns fifth_root, the echoed radicand, and a short formula string. The browser tool is free.
What does the fifth root calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.