# Simplify Radicals Calculator - Rationalize, With Steps

> Simplify square, cube and nth roots (√72 = 6√2, ∛54 = 3∛2) with steps, rationalize denominators, add and multiply radicals, and get roots to 1000 digits.

- URL: https://qilingtools.com/en/math/radical
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Simplest radical form is built on prime factorization: to simplify √72, write 72 = 2³ × 3², pull every pair of equal primes out of the root and you get 6√2; likewise ∛54 = ∛(3³ × 2) = 3∛2. The Simplify tab takes an index and a radicand (an integer, fraction or decimal) and shows the factorization, how the factors are grouped and pulled out, the simplified result, whether the number is a perfect square or cube, which consecutive powers it lies between, and its decimal value to as many as 1000 places. The Expression tab evaluates whole expressions exactly — √8 + √18, 2√3 × 3√6, 1/(√3 − 1), 1/(1 + √2 + √3), 1/(∛2 + 1) — combining like radicals, multiplying and dividing, and rationalizing denominators step by step (conjugates for square roots, the sum-of-cubes identity for cube roots). It even denests roots such as √(3 + 2√2) = 1 + √2.

## How to use

1. On the Simplify tab, enter the index (2 for a square root) and the radicand, e.g. 72, 54 or 3/4, or pick an example
2. Read the prime factorization, the grouping step and the simplest radical form; choose how many decimal places you want
3. Switch to the Expression tab to evaluate things like √8 + √18 or 1/(√3 − 1) and see the exact answer with rationalization steps

## Features

- Simplifies square, cube and any nth roots with prime-factorization steps
- Fraction, decimal and negative radicands (odd roots)
- Perfect square / cube test and the bracketing powers, e.g. 8² < 72 < 9²
- Exact root digits to 1000 places using integer arithmetic
- Exact radical arithmetic: combine like radicals, multiply, divide, raise to powers
- Rationalizes monomial, multi-term square-root and binomial cube-root denominators
- Denests roots like √(3 + 2√2) = 1 + √2
- Accepts √ ∛ ∜, sqrt, cbrt, root(x, n) and coefficients written as 2√3

## FAQ

### How do you simplify a square root?

Factor the radicand into primes and move each pair of equal primes outside the root as a single factor. 72 = 2³ × 3² = (2 × 3)² × 2, so √72 = 6√2; 200 = 10² × 2, so √200 = 10√2. A radical is in simplest form when the radicand has no perfect-square factor and no fraction. Cube roots work the same way with groups of three: ∛54 = ∛(3³ × 2) = 3∛2.

### How do you rationalize a denominator?

For a single root, multiply top and bottom by that root: 1/√2 = √2/2. For a binomial a + √b, multiply by the conjugate a − √b so the difference of squares removes the root: 1/(√3 − 1) = (√3 + 1)/(3 − 1) = (1 + √3)/2. With three terms like 1 + √2 + √3, clear √2 with one conjugate and then √3 with another. For ∛2 + 1, use the sum of cubes a³ + b³ = (a + b)(a² − ab + b²). The calculator lists every factor it multiplies by.

### How do you add and multiply radicals?

Simplify each radical first; radicals with the same index and radicand are like radicals and their coefficients add: √8 + √18 = 2√2 + 3√2 = 5√2, and √12 − √27 + √75 = 4√3. To multiply, multiply the radicands and simplify: 2√3 × 3√6 = 6√18 = 18√2. Roots with different indices are rewritten with a common index first, e.g. √2 × ∛2 = ⁶√8 × ⁶√4 = ⁶√32.

### When does the calculator give only a decimal?

Not every expression can be written as a finite sum of radicals. A nested root that does not denest, like √(2 + √3 + √5), or a denominator with several different cube roots, is reported as not exactly representable and you get a 12-significant-digit approximation instead. Radicands above 10¹³ cannot be factored quickly and are also flagged. The decimal value of a single root is always exact to the digits shown (truncated).

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