How this calculation works
Factor the absolute radicand and group identical prime factors in pairs for a square root or triples for a cube root. Each complete group contributes one factor outside the radical. Remaining factors stay inside. Perfect powers leave no radical. Zero and negative real cube roots are handled explicitly.
If n = aᵏb, then the real kth root is a × kth-root(b), subject to the selected root’s sign ruleA worked example
72 = 36×2, so √72 = 6√2, approximately 8.48528137424. For the cube root, 72 = 8×9, giving 2∛9. The real cube root of −27 is −3.
Good to know
Whole-number radicands from −10¹² to 10¹². Negative square roots are rejected in this real-only tool; negative cube roots are allowed. Exact radical simplification does not imply the displayed decimal is exact. No symbolic variables, rational radicands or complex-root branches.
Reference: OpenStax Prealgebra: fractions and representations (opens in a new tab) ↗
Perfect powers leave the radical in complete groups
For a square root, every pair of identical prime factors supplies one factor outside. For a cube root, the corresponding group has three factors. The factors left inside explain why one integer can simplify differently under the two root choices.
For 72, square-root grouping extracts 36 and leaves two, giving 6√2. Cube-root grouping extracts eight and leaves nine, giving 2∛9. The square root calculator and cube root calculator provide a direct numeric view; this page preserves the simplified radical form alongside that approximation.
The decomposition is checked by reversing it: if the extracted factor is a and the remaining radicand is b, then a²b or a³b must reproduce the original radicand under the selected root convention. Perfect powers leave b equal to one.
Root rules need their sign and operation conditions
The principal real square root is nonnegative, so √9 is three rather than both signs. Solving x²=9 is a different task whose solutions are ±3. A negative radicand has no real square root, whereas a real cube root can be negative: ∛(−27)=−3.
A root does not distribute over addition. For example, √(9+16)=5, while √9+√16=7. Factoring a product into a perfect power and a remainder is a valid simplification here; splitting a sum into separate roots is not.
An integer radical may be exact even when its decimal expansion does not terminate. The exact simplified expression and the approximate numeric result serve different purposes. Keep the radical form when later exact symbolic work depends on it.
A few questions, answered
Why does √9 show only 3?
This tool returns the principal square root. The equation x²=9 is a different problem with two roots.
Can I simplify a negative cube root?
Yes. The real cube root preserves the negative sign.
Does simplifying a radical make its decimal exact?
No. An irrational radical still has a nonterminating decimal representation.