Simplify radical
Rewrite a square root in simplest radical form: √12 becomes 2√3 and √50 becomes 5√2, with the number that comes out and the part that stays under the root shown separately. Works for whole numbers up to one trillion. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 12
- Coefficient
- 2
- Radicand
- 3
- Simplified
- 2√3
- Expression
- √12 = 2√3
Simplest radical form is the algebra habit of rewriting a square root so that the integer under the radical no longer hides a perfect-square factor. Students meet it when √12 becomes 2√3, √50 becomes 5√2, or √72 becomes 6√2, and they lose marks when a leftover four or nine stays inside the root. This simplify-radical calculator accepts one positive integer n, extracts the largest square factor, and returns the coefficient that comes out together with the square-free radicand that stays under the radical. Failed validation is not billed.
What simplest radical form means and how the coefficient comes out
A square root of a positive integer n is already in simplest radical form when n has no perfect-square factor greater than one. The identity behind every textbook drill is √(a²b) equals a√b whenever a and b are positive integers and b is square-free. Finding that a is the same as pulling the largest integer whose square still divides n: divide n by that square, keep the quotient as the remaining radicand, and the integer you pulled is the coefficient.
Classic checks include √8 equals 2√2, √12 equals 2√3, √18 equals 3√2, √32 equals 4√2, √50 equals 5√2, and √72 equals 6√2. When n itself is a perfect square the remaining radicand is one and the simplified form is just the integer root: √16 equals 4, √36 equals 6, √100 equals 10. When n is already square-free the coefficient is one and the radical stays as √n, so √7, √11, and √15 are already finished.
Nothing is approximated with a floating-point square root. The factorisation walks exact integer squares, so a near miss such as 48 never becomes a false perfect square and never leaves a four sitting under the radical.
Where a simplify-radical calculator earns its keep
Classrooms and tutoring platforms use a simplify-radical calculator so learners can check extracted coefficients for arbitrary positive integers without flipping a factor tree by hand and missing a square. Exam-prep generators need a deterministic oracle that marks leftover square factors wrong and that rejects zero or a negative as invalid input instead of silently returning a complex root. Worksheet builders and learning-management plugins can mint hundreds of √n items and store the official coefficient and remaining radicand beside each stem.
Failed validation does not consume balance. Keep requests small, pin the returned coefficient and radicand in tests, and you get reproducible simplest-radical answers for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom and homework checks
Let students submit a positive integer n and verify that √n reduces to the official coefficient and square-free radicand, including the integer answer when n is a perfect square.
Exam and LMS auto-grading
Mark simplest-radical items with a deterministic oracle that rejects zero and negatives as invalid input and never leaves a hidden square under the root.
Worksheet and content generation
Mint √n drill items and store the simplified coefficient plus remaining radicand beside each stem without importing a computer-algebra library.
FAQ
What happens if n is not a positive integer?
Zero, negatives, fractions, and missing n are rejected and are not billed.
What does the calculator return for a perfect square?
For example √16 returns coefficient 4 and simplified 4.
Does the engine use a floating-point square root?
No. It extracts square factors with exact integer arithmetic so the leftover radicand is square-free and goldens compare byte for byte.
How do I simplify the square root of 72?
Find the largest perfect square that divides 72, which is 36. Its root, 6, comes out in front and 2 stays under the radical, so √72 becomes 6√2.
How do I know a radical is fully simplified?
When the number left under the root has no perfect-square factor other than one. √7, √11 and √15 are already finished, while √48 is not: it still hides a square and simplifies to 4√3.
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/radical \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":12}'const res = await fetch("https://api.kit.forhosting.com/math/radical", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 12
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/radical",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 12
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/radical", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":12}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":12}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/radical", 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": 12
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.radical",
"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.
Limits
max_n | 1000000000000 |
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
The same deterministic module powers the free browser tool and the API, so a homework check and a grading script never disagree on the same integer. Successful API calls use the published $0.002 base price with no model tokens and no outbound network.
What simplest radical form means and how the coefficient comes out
This capability encodes exactly that contract: field n is the original radicand, coefficient is the extracted integer, radicand is the square-free remainder, is_perfect_square is true only when that remainder is one, simplified is the compact classroom string, and expression restates √n equals that string.
Input contract, algorithm, and how errors are reported
The engine factors n by extracting every square i squared that still divides the remaining value, first peeling factors of four and then odd trial integers, using exact BigInt arithmetic so products stay precise up to the published limit. Example: n equals seventy-two yields coefficient six and radicand two. Failed validation does not consume balance. Keep n inside the declared limit; a radicand larger than one trillion is rejected rather than allowed to run an unbounded factor loop. POST a JSON body with the required field n, a positive integer from one through one trillion. The same value is accepted under the aliases value, number, and radicand so forms and query strings can reuse one contract. Each may arrive as a JSON integer or as a whole decimal string; optional leading plus is allowed; fractions, embedded spaces, scientific notation in strings, leading zeros, zero itself, negatives, non-finite numbers, and missing fields all fail with invalid_input naming n. On success the body echoes n and returns coefficient, radicand, is_perfect_square, simplified, and expression. Example: n equals twelve yields coefficient two, radicand three, is_perfect_square false, simplified 2√3, and expression √12 = 2√3. Example: n equals sixteen yields coefficient four, radicand one, is_perfect_square true, and simplified 4. Pair the response with your own typesetting if you need a rendered radical glyph; this endpoint returns the numbers and a plain-text expression only.
Where a simplify-radical calculator earns its keep
Engineering and science notebooks sometimes reduce nested radicals that appear after a Pythagorean step or a variance formula to a square-free radicand before they display a result; calling a tiny parse API avoids pulling a full computer-algebra system for a single integer.
Technical questions
How do I simplify a square root radical √n?
Send a positive integer n (aliases value, number, radicand). The API returns coefficient and radicand so that √n = coefficient × √radicand, plus a compact simplified string such as 2√3.
What does the simplify-radical calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.