Simplify radical expressions by extracting perfect square, cube, or nth-root factors. See prime factorization, step-by-step extraction, and visual grouping of prime factors.
The Simplify Radicals Calculator reduces any radical expression to its simplest form by extracting perfect square, cube, or higher-power factors from under the radical sign. Simplifying radicals is a foundational algebra skill used throughout mathematics — from basic equation solving to calculus and beyond.
The process works by prime factorizing the radicand (the number under the radical), grouping prime factors into sets equal to the root index, and extracting one factor from each complete group. For a square root, pairs of identical primes come out; for a cube root, triples come out. The calculator visualizes this grouping with color-coded blocks, making it immediately clear which factors get extracted and which remain.
Enter any radicand and choose the root index (square root, cube root, fourth root, or fifth root). You can also include a coefficient that multiplies the radical. The calculator displays the complete prime factorization table, the number of extractable groups for each prime, and the final simplified form. Eight presets cover common textbook values, and an optional comparison field lets you simplify a second radical side by side. A perfect powers reference table helps you recognize perfect squares, cubes, and fourth powers up to 12⁴ at a glance.
Simplify Radicals Calculator helps you solve simplify radicals problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Radicand (number under radical), Index (2=square root, 3=cube root), Coefficient (multiplier outside radical) once and immediately inspect Original Expression, Simplified Form, Decimal Value to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
For ⁿ√(p₁^a₁ · p₂^a₂ · …), extract pᵢ^⌊aᵢ/n⌋ from each prime and leave pᵢ^(aᵢ mod n) inside. Result: (∏ pᵢ^⌊aᵢ/n⌋) · ⁿ√(∏ pᵢ^(aᵢ mod n)).
Result: Original Expression shown by the calculator
Using the preset "√72", the calculator evaluates the simplify radicals setup, applies the selected algebra rules, and reports Original Expression with supporting checks so you can verify each transformation.
This calculator takes Radicand (number under radical), Index (2=square root, 3=cube root), Coefficient (multiplier outside radical), Show Prime Tree and applies the relevant simplify radicals relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Original Expression, Simplified Form, Decimal Value, Extracted Factor to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It means rewriting the expression so no perfect square (or cube, etc.) factors remain under the radical sign. For example, √72 = 6√2 because 72 = 36 × 2 and √36 = 6.
A radical ⁿ√k is fully simplified when no prime factor of k has an exponent ≥ n. For square roots, no prime factor should appear more than once under the radical.
Yes — the same principle applies. For √(x⁴y³), extract x² and y¹ to get x²y√y. This calculator handles numeric radicands.
√ is the square root (index 2) — you extract pairs of factors. ∛ is the cube root (index 3) — you extract triples of factors.
Simplified radicals are easier to compare, combine, and use in further calculations. It is also standard mathematical convention, required on tests and in published solutions.
For odd-index roots (cube root, fifth root), yes — ∛(−8) = −2. For even-index roots (square root, fourth root), negative radicands produce non-real (complex) results.