This simplify radicals calculator reduces square, cube and higher roots to simplest radical form — perfect powers factored out of the radicand — with the decimal value alongside.

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Pull the Perfect Powers Out From Under the Root

The root functions themselves: our Square Root and Cube Root Calculators. Part of our Math Calculators collection.

Worked Examples (Verified)

  • √72 (default) = 6√2 ≈ 8.485 — because 72 = 36 × 2
  • √180 = 6√5: 180 = 36 × 5; the largest perfect square wins in one step (chasing 4 × 45 = 2√45 = 2·3√5 also lands here, slower)
  • ∛54 (index 3, radicand 54) = 3∛2: cubes this time — 54 = 27 × 2. And √48 = 4√3 (48 = 16 × 3)

Simplest Radical Form Rules

  • No perfect nth power left under an nth root (for squares: no factor 4, 9, 16, 25…).
  • No fractions under the root, and no roots in denominators (rationalize those separately).
  • Even roots of negatives aren't real; odd roots of negatives are fine and keep the sign (∛−54 = −3∛2).
  • Like radicals (same index, same radicand) combine like variables: 6√2 + 3√2 = 9√2 — simplification is what makes them visibly “like”.

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Frequently Asked Questions

How do you simplify a radical?

Factor the radicand, pull out every group of factors matching the root's index (pairs for square roots, triples for cube roots), and leave the rest inside. Example: √72 = √(36 × 2) = 6√2. This calculator finds the largest perfect power automatically, for any index.

What is simplest radical form?

The form where the radicand contains no perfect-power factor for the root's index, no fraction sits under the root, and no root remains in a denominator. 6√2 is simplest; √72, 2√18 and 3√8 are all the same number not yet simplified.

Can cube roots of negative numbers be simplified?

Yes — odd roots preserve sign: ∛−54 = −3∛2. Even roots of negatives (√−16) have no real value, and this calculator says so rather than returning a numeric error.

Why simplify radicals at all in the calculator era?

Exactness: 6√2 is the number; 8.485 is a rounding of it. Algebra downstream (adding like radicals, exact geometry answers, Pythagorean results like √50 = 5√2) works in the exact form — decimals dissolve structure, radicals preserve it.

How do you simplify √180?

Find the largest perfect square dividing 180: 36 × 5 → √180 = 6√5 ≈ 13.416. If you spot only 4 × 45 first, repeat: 2√45 = 2√(9×5) = 6√5 — simplifying is idempotent; every path ends at the same simplest form.

Is this simplify radicals calculator free?

Yes — free, instant, no sign-up.