Math July 13, 2026 · 8 Min Read

Simplifying Radicals Calculator – Guide & Formulas

Simplify square roots, cube roots, and nth roots to simplest radical form. See step-by-step factor extraction and final simplified result.

Simplify any radical to its simplest form with our free online calculator. Enter a square root, cube root, or nth root to see the simplified radical and step-by-step factor extraction.

Key Takeaway

Use the free Simplifying Radicals Calculator to simplify square roots, cube roots, and nth roots to simplest radical form. see step-by-step factor extraction and final simplified result. Get instant results with step-by-step explanations.

How to Use the Simplifying Radicals Calculator

  1. Step 1: Enter the number inside the radical (the radicand)
  2. Step 2: Enter the root index (2 for square root, 3 for cube root, etc.)
  3. Step 3: Click Simplify to see the simplest radical form
  4. Step 4: Review the prime factorization and step-by-step extraction

The Formula

To simplify an nth root: factor the radicand, group identical prime factors in sets of n, and extract each group as a single factor outside the radical. √(a × b²) = b√a for square roots.

Variable Definitions

  • √: The radical symbol — denotes a root operation
  • n: The index of the root (2 for square root, 3 for cube root, etc.)
  • Radicand: The number inside the radical symbol
  • Simplest form: The radical with no perfect nth power factors remaining inside

Simplifying √72

Reduce the square root of 72 to its simplest radical form.

  1. Step 1: Find the prime factorization of 72: 72 = 2³ × 3² = 8 × 9
  2. Step 2: Identify perfect square factors: 2² = 4 and 3² = 9 are perfect squares
  3. Step 3: Extract perfect squares: √(36 × 2) = √36 × √2 = 6√2
  4. Step 4: Simplified form: 6√2 (the remaining 2 has no perfect square factors)

Frequently Asked Questions

How do I simplify a square root?

Find the prime factorization of the radicand. Group identical factors in pairs. For each pair, extract one factor outside the radical. Leave any unpaired factors inside.

What is the simplest radical form?

Simplest radical form means the radicand has no perfect nth power factors (for the given index n). For square roots, no perfect square factors remain inside.

Can I simplify radicals with variables?

Yes. For √(x⁴y³), extract x² (from x⁴) and y (from y³) outside: x²y√y. Variables with exponents ≥ n (the root index) can be extracted.

How do I simplify cube roots?

Group identical prime factors in sets of three. For each group of three, extract one factor. For example, ∛54 = ∛(27 × 2) = 3∛2.

What if the radicand is a fraction?

Simplify numerator and denominator separately. √(a/b) = √a / √b. Then rationalize the denominator if needed by multiplying top and bottom by √b.

How do I combine simplified radicals?

Only radicals with the same index and same radicand can be combined. For example, 3√2 + 5√2 = 8√2, but 3√2 + 3√3 cannot be simplified further.