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Dividing radicals calculator

√8 divided by √2 is exactly 2, and √8 over √3 simplifies to 2√6/3. Enter each radical's number and root index to get the quotient fully simplified, with the denominator already rationalized. Free in your browser.

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Worked example

If you enter

Radicand a: 8
Index a: 2
Radicand b: 2
Index b: 2

You get

Coefficient: 2
Index: 1
Radicand: 1
Expression: 2

Dividing one radical by another is the quotient rule for roots: the n-th root of a over the m-th root of b is rewritten with a common index, simplified by extracting perfect powers, and then rationalized so no radical remains in the denominator. This dividing radicals calculator accepts the radicand and the root index of the dividend and of the divisor, applies the identity that the n-th root of a divided by the m-th root of b equals the L-th root of a to the power L over n over b to the power L over m, where L is the least common multiple of the two indices, extracts every perfect L-th power as a rational coefficient, and multiplies numerator and denominator by the leftover root that clears the denominator. The response is the simplified coefficient, remaining index, and radicand, plus a display string and a numeric quotient you can pin in tests. An index that is not a positive integer is rejected the same way.

What the quotient of two radicals means and how it is rationalized

A radical ⁿ√a is the principal real n-th root of the radicand a, provided a is non-negative and n is a positive integer. Dividing one such root by another is the quotient rule for radicals. When the indices already match, ⁿ√a divided by ⁿ√b equals ⁿ√(a/b), after which you simplify the single resulting root and, if a radical would remain under a denominator, multiply numerator and denominator by a leftover root that clears that denominator.

When the indices differ, rewrite both factors with L equal to the least common multiple of n and m: ⁿ√a becomes ᴸ√(a raised to L over n) and ᵐ√b becomes ᴸ√(b raised to L over m), so the quotient is the L-th root of a to that first power over b to that second power. Simplifying then means peeling every perfect L-th power out as a rational coefficient and leaving an L-power-free radicand in the numerator.

Rationalizing is the extra step that a product of roots does not need: leftover prime exponents that would sit in the denominator are lifted into the numerator by multiplying by the complementary power, so a textbook result such as the square root of eight over the square root of three becomes two times the square root of six, all over three, never an unreduced radical sitting under a fraction bar.

When the leftover radicand is one, the quotient is already rational and the calculator returns index one with radicand one, so clients can treat a fully simplified value such as two as a plain coefficient. Zero as a dividend is a legal zero quotient; zero as a divisor is undefined and is rejected rather than returning Infinity.

Where dividing radicals comes up

Classrooms and tutoring platforms use a dividing radicals calculator so learners can check the quotient rule and the rationalization of a leftover root without dropping a factor of two when they rewrite the square root of eight over the square root of three. Exam-prep generators need a deterministic oracle that marks a negative radicand or a non-integer index as invalid instead of silently returning NaN that a buggy grader might treat as success. Failed validation does not consume balance.

The same contract also covers mixed indices such as a cube root over a square root, which the common-index rewrite and rationalization handle in one shot.

What you can do with it

Classroom and homework checks

Let students enter two radicands and indices and verify the simplified, rationalized quotient against the common-index method used in textbooks.

Exam and LMS auto-grading

Mark dividing-radicals items with a deterministic oracle that rejects a negative radicand or a non-integer index and never invents a complex root.

Notebooks that reduce nested root quotients

Divide and rationalize two real radicals in calibration or geometry scripts without importing a full computer-algebra library.

FAQ

How do I divide two radicals with the same index?

Send the two radicands and the shared index. The quotient is the same-index root of the quotient of the radicands, then simplified and rationalized (for example √8 / √2 = √4 = 2, and √8 / √3 = 2√6/3).

How do I divide radicals that have different indices?

Send both indices. The calculator rewrites the factors with L = lcm(n, m) so ⁿ√a / ᵐ√b = ᴸ√(a^{L/n} / b^{L/m}), then extracts perfect L-th powers and rationalizes the denominator.

Why is a negative radicand rejected?

This calculator divides non-negative real radicals only. An index that is not a positive integer is rejected the same way.

What does rationalizing the result mean?

Leftover roots are moved out of the denominator: √8 / √3 becomes 2√6/3, not an unreduced √(8/3). A fully rational quotient uses index 1 and radicand 1. The response sets rationalized to true.

Which roots can I divide with this calculator?

Any root with a whole-number index from 2 to 24, where 2 is a square root and 3 is a cube root. The numbers under the roots must not be negative, and the divisor cannot be zero.

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

POSThttps://api.kit.forhosting.com/math/dividing-radicals

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/dividing-radicals \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"radicand_a":8,"index_a":2,"radicand_b":2,"index_b":2}'

const res = await fetch("https://api.kit.forhosting.com/math/dividing-radicals", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "radicand_a": 8,
    "index_a": 2,
    "radicand_b": 2,
    "index_b": 2
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/math/dividing-radicals",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "radicand_a": 8,
    "index_a": 2,
    "radicand_b": 2,
    "index_b": 2
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/dividing-radicals", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"radicand_a":8,"index_a":2,"radicand_b":2,"index_b":2}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"radicand_a":8,"index_a":2,"radicand_b":2,"index_b":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/dividing-radicals", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "radicand_a": 8,
  "index_a": 2,
  "radicand_b": 2,
  "index_b": 2
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "math.dividing_radicals",
  "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.

Limits

`max_abs_radicand`	1000000000000
`min_index`	2
`max_index`	24

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 not accepted: this tool works over the non-negative reals and fails with invalid_input instead of returning a complex placeholder.

Input contract, response shape, and error behaviour

Indices must be positive integers from two through twenty-four; two is a square root and three is a cube root. A negative radicand fails with a clear message that this calculator only divides non-negative real radicals; an index that is not a positive integer in the published range fails the same way. A zero divisor is undefined and is also rejected. Those failures are not billed. POST a JSON body with four required finite numbers: radicand_a and index_a for the dividend radical, radicand_b and index_b for the divisor radical. Each radicand may arrive as a JSON number or as a numeric string, including a rational that converts exactly to a fraction, and both radicands must be non-negative. Aliases a and n for the dividend and b and m for the divisor are accepted so scripts can reuse common textbook names. Missing fields, empty strings, and non-finite values fail with invalid_input naming the field. On success the body echoes the four inputs, returns coefficient, index, and radicand of the simplified rationalized quotient, an expression string such as 2 or 2√6/3, an unsimplified display of the original quotient, a quotient_value rounded to twelve decimal places for stable goldens, a rationalized flag that is true once the denominator is free of leftover roots, and a formula string restating the common-index identity. Example: radicand_a eight, index_a two, radicand_b two, index_b two is the square root of eight divided by the square root of two, which simplifies to two. The same four fields with radicand_b three yield two times the square root of six over three after rationalizing.

Where a dividing-radicals API earns its keep

Science and engineering notebooks often reduce nested roots that appear after dividing characteristic lengths, impedances, or wave amplitudes; calling a tiny parse endpoint avoids pulling a full computer-algebra system for two indices and two radicands. Keep requests inside the published magnitude and index limits, pin the rounded quotient_value and the expression string in tests, and you get reproducible simplified quotients for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.

Technical questions

What does the dividing radicals calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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