Softonic

Radical Simplifier

Step 1 / 2 Results
Results

A square root is simplified when no perfect-square factor remains inside it. Enter a non-negative integer up to 1 billion and the tool returns an exact coefficient-and-radical form plus a decimal approximation.

How to simplify a radical

  1. 1

    Enter the radicand

    Type the integer under the root, e.g. 72 for sqrt(72).

  2. 2

    Perfect squares are extracted

    The calculation removes square factors internally; for example, 72 contains 36 x 2.

  3. 3

    Pair up factors

    Every pair leaves the root as a single copy; 2x2 becomes 2 outside, 3x3 becomes 3 outside.

  4. 4

    Read both results

    For 72, the exact result is 6√2 and the decimal approximation is about 8.48528137.

Worked examples

Original Factored Simplified
sqrt(8) sqrt(4 x 2) 2 sqrt(2)
sqrt(50) sqrt(25 x 2) 5 sqrt(2)
sqrt(72) sqrt(36 x 2) 6 sqrt(2)
sqrt(98) sqrt(49 x 2) 7 sqrt(2)
sqrt(180) sqrt(36 x 5) 6 sqrt(5)
sqrt(128) sqrt(64 x 2) 8 sqrt(2)

The algorithm

  1. Factor the radicand into primes.
  2. Group identical primes into pairs.
  3. Move each pair out of the root as a single copy.
  4. Multiply the outside factors; keep the remaining primes inside the root.
  5. If nothing remains inside, the result is a plain integer.

Tip for cube roots and higher

The same idea works for cube roots (group factors in threes), fourth roots (groups of four) and so on. This tool simplifies square roots only, but the principle generalises.

Frequently Asked Questions

Simplified form is exact; decimal form is rounded. Teachers and physics graders mark 6 sqrt(2) as correct and 8.485 as “approximately”. Calculus identities also work better in simplified form.

sqrt of a negative is imaginary. This tool handles real simplifications only; feed it non-negative integers.

Yes. sqrt(0) is 0. Nothing to factor.

No. The input accepts whole numbers only. You can simplify sqrt(1/2) separately as sqrt(2)/2, but this tool does not parse fractions or expressions.

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