Math July 13, 2026 · 8 Min Read

Radical Calculator – Guide & Formulas

Simplify square roots, cube roots, and nth roots. Convert radicals to simplest form, find perfect square factors, and evaluate with step-by-step solutions.

Simplify any radical expression with our free online Radical Calculator. Enter square roots, cube roots, or nth roots to see simplest form, perfect factors, and step-by-step simplification.

Key Takeaway

Use the free Radical Calculator to simplify square roots, cube roots, and nth roots. convert radicals to simplest form, find perfect square factors, and evaluate with step-by-step solutions. Get instant results with step-by-step explanations.

How to Use the Radical Calculator

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

The Formula

Simplifying radicals: Extract the largest perfect nth power factor from the radicand. √(a) = √(b² × r) = b√(r), where b² is the largest perfect square dividing a.

Variable Definitions

  • √: The radical symbol — denotes a root
  • n: The index of the root (2 for square root, 3 for cube root, etc.)
  • a: The radicand — the number inside the radical
  • b: The largest integer whose nth power divides the radicand
  • r: The remaining radicand after extracting perfect powers

Simplifying √72

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

  1. Step 1: Find prime factorization of 72: 72 = 2³ × 3² = 8 × 9
  2. Step 2: Identify perfect square factors: 72 = 36 × 2 = 6² × 2
  3. Step 3: Extract the perfect square: √72 = √(36 × 2) = 6√2
  4. Step 4: Verify: 6√2 ≈ 6 × 1.4142 ≈ 8.485 ≈ √72 ✓

Frequently Asked Questions

How do I simplify cube roots?

Find the largest perfect cube factor of the radicand. For example, ∛54 = ∛(27 × 2) = 3∛2 because 27 is the largest perfect cube dividing 54.

What is the difference between a perfect square and a radical?

A perfect square is a number whose square root is an integer (e.g., 25 = 5²). A radical is an irrational root that cannot be simplified to a whole number (e.g., √2).

Can I add or subtract different radicals?

You can only add or subtract radicals with the same radicand and index. For example, 3√2 + 5√2 = 8√2, but you cannot directly add √2 + √3.

How do I rationalize a denominator?

Multiply both numerator and denominator by the radical to eliminate the root from the denominator. For example, 1/√2 becomes √2/2 after rationalizing.

What are nth roots?

The nth root of a number a is a value that, when raised to the nth power, equals a. For example, ∛8 = 2 because 2³ = 8.