Simplify Radical Calculator
Reduce a square root or nth root to its exact simplest radical form. Enter the number under the radical (and optionally a different root index) to see every prime factor pulled out and what stays inside.
What does it mean to simplify a radical?
Simplifying a radical means rewriting it so that no perfect power of the root's index is left under the radical sign. For a square root, you pull out any perfect-square factor; for a cube root, any perfect-cube factor; and so on. The value doesn't change — you're just splitting the radical into an integer part in front and a smaller radical behind it.
How the simplification works, piece by piece
Each card explains one thing this tool does with your radicand — the product rule that lets factors escape, the grouping trick that works for any root index, and how it decides a radical is already as simple as it gets.
Product rule — split off perfect powers
The radical of a product is the product of the radicals, so any perfect n-th power factor comes out as an integer coefficient in front of a smaller radical.
Prime factors — group in n's for the n-th root
Factor the radicand into primes, group them in sets of n (the index), and pull one copy of each complete group out. Loose primes stay inside.
When it can't be simplified further
A radical is already in simplest form when no prime appears at least n times in the factorization — there is no complete group to pull out.
Features of this calculator
- Simplifies square roots and any nth root (index ≥ 2)
- Returns exact radical form, not just a decimal approximation
- Shows the full prime factorization of the radicand
- Pulls out every perfect nth-power factor and keeps the rest inside
- Includes the decimal approximation as a check
Frequently asked questions
+How do I simplify a square root?
Prime-factorize the number, pair up matching primes, and pull one copy of each pair outside the radical. Whatever doesn't pair up stays inside.
+What is a perfect square factor?
A factor that is itself a perfect square — 4, 9, 16, 25, 36, 49, 64, … When the number under a square root has one of these as a factor, you can pull its root out front.
+Does this work for cube roots?
Yes — set the root index to 3. The calculator groups prime factors into sets of three instead of pairs. For any index n it groups them into sets of n.
+Why is √8 = 2√2?
Because 8 = 4 · 2 and 4 is a perfect square (2²). So √8 = √4 · √2 = 2√2. Prime-factor form: 8 = 2³ = 2² · 2, one pair of 2's comes out and one 2 stays under.