Rationalize denominator calculator
Clear the square root from the bottom of a fraction: 1/√8 becomes √2/4, with the square factor pulled out and the result reduced, and 6/√9 simplifies all the way to 2. Works with 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
- Numerator
- 1
- Radicand
- 8
- Value
- 0.3535533906
- Original
- 1/√8
- Result coefficient
- 1
- Result radicand
- 2
Fractions that hide a square root in the denominator show up on every algebra worksheet, entrance exam, and physics homework set that still asks students to rationalize before they simplify. People type rationalize the denominator when they have a over square root of n and need the matching form with a rational bottom, not a computer-algebra dump. This tool accepts an integer numerator and a non-negative integer radicand, multiplies top and bottom by the same square root, pulls every perfect-square factor out of the leftover radical, and reduces the resulting rational coefficient. Zero as a radicand is rejected as division by zero.
What rationalizing the denominator means and how the rewrite works
Rationalizing a denominator is the rewrite that turns a fraction whose bottom still contains a square root into an equivalent fraction whose bottom is a rational number. For an input of the form a over square root of n, the identity is multiply numerator and denominator by the same square root: a over square root of n equals a times square root of n, all over n. That already makes the denominator rational, but textbooks also require the leftover radical to be square-free and the rational coefficient to be reduced.
The algorithm therefore factors the radicand, writes n as k squared times m with m square-free, replaces square root of n by k times square root of m, and simplifies a over k square root of m into a times square root of m over k times m. Greatest-common-divisor reduction then cancels any common factor of a and of k times m, so one over square root of eight becomes square root of two over four rather than an unreduced two square root of two over eight.
When the radicand is a perfect square the leftover m is one and the result is fully rational: six over square root of nine is two. When the numerator is zero the value is zero regardless of the positive radicand. A negative radicand has no real square root, so the capability refuses it instead of returning a complex result that a secondary-school worksheet does not expect.
The display strings keep the radical glyph so a page can show both the original a over square root of n and the rationalized form without a typesetting library of its own.
Where a rationalize-the-denominator calculator earns its keep
Classrooms and tutoring platforms use a rationalize the denominator calculator so learners can check the rewrite of a over square root of n without dropping a perfect-square factor or forgetting to reduce. Exam-prep generators need a deterministic oracle that marks a negative radicand as invalid instead of inventing a complex answer a grader might treat as success. Construction and shop drawings sometimes keep exact surds beside a cut list so a later rounding step does not drift; feeding those pairs through the same rewrite keeps the printed form consistent.
Failed validation does not consume balance. Keep requests inside the published integer bound, pin the reduced coefficient and leftover radicand in tests, and you get reproducible rationalized fractions 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 numerator and a radicand for a/√n and verify the reduced rationalized form, including leftover square-free radicals, against the textbook method.
Exam and LMS auto-grading
Mark rationalize-the-denominator items with a deterministic oracle that rejects a negative radicand as no real value and never invents a complex result.
Notebooks and exact amplitudes
Rewrite 1/√n style amplitudes in physics or engineering notes into an exact rationalized form next to the decimal, without importing a computer-algebra library.
FAQ
How do I rationalize a denominator of the form a/√n?
Send integer fields numerator (a) and radicand (n > 0).
What if the radicand is a perfect square?
The leftover radical is 1 and the result is fully rational.
Does the calculator simplify square factors inside the radical?
Yes. 1/√8 is returned as √2/4, not as √8/8. Perfect-square factors are pulled out and the remaining coefficient is reduced by gcd.
What happens if the number under the root is negative or zero?
Neither gives a result. A negative number has no real square root, and a zero under the root would mean dividing by zero, so the calculator refuses both instead of returning a complex or undefined answer.
Why is 1/√8 written as √2/4 and not 2√2/8?
Because the result is fully reduced: √8 is first rewritten as 2√2, and the common factor between the top and the bottom is cancelled. Both forms are equal, but textbooks expect the reduced one, √2/4.
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/rationalize-denominator \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"numerator":1,"radicand":8}'const res = await fetch("https://api.kit.forhosting.com/math/rationalize-denominator", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"numerator": 1,
"radicand": 8
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/rationalize-denominator",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"numerator": 1,
"radicand": 8
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/rationalize-denominator", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"numerator":1,"radicand":8}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"numerator":1,"radicand":8}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/rationalize-denominator", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"numerator": 1,
"radicand": 8
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.rationalize_denominator",
"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_abs | 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
A negative radicand is rejected as invalid_input because there is no real square root to place in a real denominator.
Input contract, response shape, and error behaviour
Radicand may also be sent as rad, n, or value. A negative radicand is rejected because the square root of a negative number is not real, which is the product contract this capability advertises. A radicand of zero is rejected as division by zero: a over square root of zero is undefined. Example: numerator 1 and radicand 8 yields coefficient 1, leftover 2, denominator 4, and rationalized √2/4. Example: numerator 6 and radicand 9 yields the fully rational integer 2. POST a JSON body with two required integers: numerator, the integer a on top of the original fraction, and radicand, the integer n under the square root in the denominator. Numerator may arrive as a JSON integer or as a signed digit string; aliases num and a are accepted when numerator is omitted. Missing fields, empty strings, non-integer text, NaN, and Infinity fail with invalid_input naming the field and are not billed. Magnitudes above one trillion are rejected so every returned integer stays inside the safe JSON number range. On success the body echoes the parsed numerator and radicand; repeats the original as a display string such as 1/√8; lists result_coefficient, result_radicand, and result_denominator as the reduced parts of coefficient times square root of leftover over a rational bottom; sets rationalized to a display string such as √2/4; flags is_fully_rational when the leftover radicand is one; returns value as the decimal of the rewritten form rounded to twelve places for stable comparisons; and includes a short formula string plus a one-line message. Pair the payload with your own labels; this endpoint does not emit plots or multi-step classroom prose beyond the formula and display strings.
Where a rationalize-the-denominator calculator earns its keep
Physics and engineering notebooks still rationalize expressions such as one over square root of two when they present exact amplitudes next to a decimal, and calling a tiny parse endpoint avoids pulling a full computer-algebra system for a two-integer problem.
Technical questions
What happens if the radicand is negative?
The task fails with invalid_input. There is no real square root of a negative number, so a/√n is not a real fraction and the call is not billed.
What does the rationalize denominator calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.