ForHosting KIT · Developer Utilities

Integer nth root floor calculator

The integer nth root floor calculator returns the greatest integer that does not exceed the real k-th root of n.

● BetaFree · in your browser

Use it from WebAPIEmailTelegramApp soon

It uses integer arithmetic throughout, so perfect powers and values close to a power boundary do not suffer from floating-point rounding. Positive radicands work with every positive degree, while negative radicands work with odd degrees. The result is suitable for algorithms, proofs, validation rules, and any workflow that needs a reproducible floor rather than a decimal approximation.

What the floor of an nth root means

For a nonnegative integer n and a positive integer k, the result r is characterized by two inequalities: r raised to k is at most n, while r plus one raised to k is greater than n. This definition matters because it identifies one exact integer without asking a decimal library to approximate an irrational root. For example, the cube root of 1000 is exactly 10, so the returned floor is 10. The cube root of 999 is slightly below 10, so its floor is 9. Negative values require careful use of the ordinary mathematical floor. The real cube root of -9 is a little below -2, and therefore its floor is -3, not -2. Negative radicands have real roots only for odd k in this capability; an even degree with a negative n is rejected because there is no real result to floor. Degree one is also supported and simply returns n.

How exact integer arithmetic avoids rounding errors

A common implementation computes a root with a floating-point power function and then rounds down. That shortcut can fail near exact powers, where a tiny representation error may place a computed value on the wrong side of an integer boundary. This capability instead searches for the answer using integers and compares bounded powers without converting the root to a decimal approximation. Exponentiation is performed by repeated squaring, and a multiplication stops as soon as it is known to exceed the radicand. A binary search then narrows the candidate interval until only the greatest valid integer remains. The input n is restricted to the exact safe-integer range used by JSON numbers in JavaScript, ensuring that the integer supplied by the caller has not already been rounded before the calculation starts. The degree k is bounded from 1 through 1024, which keeps runtime predictable even though large degrees usually lead quickly to roots of zero, one, or minus one.

Choosing inputs and interpreting the result

Provide n as an integer and k as the positive integer degree of the root. Use k equal to 2 for an integer square-root floor, 3 for a cube-root floor, and so on. The response repeats n and k and supplies root as the primary result, making it straightforward to preserve alongside an audit record or compare with another calculation. Remember that floor always means rounding toward negative infinity. Consequently, an inexact negative odd root becomes the next more-negative integer, whereas truncation toward zero would produce a different and incorrect answer. This distinction is useful in interval algorithms, number-theory routines, discrete geometry, and capacity planning where a root determines a whole-number bound. The calculation is deterministic, uses no network service, and retains no input. Browser and API executions share the same solving function. Automated API requests are charged $0.002 per item, while the browser execution offers the same mathematical result without requiring an approximate calculator or a hand-written correction around perfect powers.

What you can do with it

Bound a search space

Compute an exact integer root when an algorithm needs a conservative whole-number upper or lower boundary.

Test perfect-power neighborhoods

Check values immediately below or above a large power without floating-point boundary errors.

Handle negative odd roots correctly

Apply mathematical floor, rather than truncation toward zero, to negative radicands with odd degree.

FAQ

What does the capability return?

It returns the greatest integer less than or equal to the real k-th root of n, together with the supplied n and k.

Can n be negative?

Yes when k is odd. A negative n with an even k is rejected because its k-th root is not real.

Why is the floor of the cube root of -9 equal to -3?

The real cube root is approximately -2.08, and floor rounds toward negative infinity rather than toward zero.

Does it use floating-point roots?

No. It searches with exact integer comparisons, preventing rounding errors near perfect-power boundaries.

What are the input limits?

n must be an exact safe integer, and k must be an integer from 1 through 1024.

What does an API request cost?

Each item costs $0.002. The browser calculation uses the same deterministic solving logic.

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

POSThttps://api.kit.forhosting.com/numth/integer-nth-root

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/numth/integer-nth-root \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"n":1000,"k":3}'

const res = await fetch("https://api.kit.forhosting.com/numth/integer-nth-root", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "n": 1000,
    "k": 3
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/numth/integer-nth-root",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "n": 1000,
    "k": 3
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/numth/integer-nth-root", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"n":1000,"k":3}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"n":1000,"k":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/numth/integer-nth-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

{
  "n": 1000,
  "k": 3
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "numth.integer_nth_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

Per request$0.002

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.

Read the full KIT documentation →