Math July 13, 2026 · 8 Min Read

Root Calculator – Guide & Formulas

Calculate square roots, cube roots, and nth roots of any number. See exact simplified form, decimal approximation, and step-by-step solution.

Calculate any root of a number with our free online Root Calculator. Enter square roots, cube roots, or nth roots to see simplified form, decimal approximation, and steps.

Key Takeaway

Use the free Root Calculator to calculate square roots, cube roots, and nth roots of any number. see exact simplified form, decimal approximation, and step-by-step solution. Get instant results with step-by-step explanations.

How to Use the Root Calculator

  1. Step 1: Enter the number (radicand)
  2. Step 2: Enter the root index (2 for square root, 3 for cube root, etc.)
  3. Step 3: Click Calculate to see the result
  4. Step 4: Review the simplified radical form and decimal approximation

The Formula

The nth root of a is a^(1/n). The principal root is the non-negative root for even indices. For example, √9 = 3 because 3² = 9.

Variable Definitions

  • n: The root index (2 for square root, 3 for cube root, etc.)
  • a: The radicand — the number whose root we are finding
  • √: The radical symbol — denotes a root operation
  • Principal Root: The non-negative root when the index is even

Finding the Cube Root of 27

Calculate ∛27.

  1. Step 1: Set index n = 3, radicand a = 27
  2. Step 2: Ask: what number raised to the 3rd power equals 27?
  3. Step 3: Test: 3³ = 3 × 3 × 3 = 27
  4. Step 4: ∛27 = 3

Frequently Asked Questions

What is the principal root?

The principal root is the non-negative root when the index is even. For example, √9 = 3 (not -3), because 3 is the principal (non-negative) square root.

Can I take an even root of a negative number?

No, the even root of a negative number is not a real number. For example, √(-4) is undefined in real numbers but equals 2i in complex numbers.

What is the relationship between roots and exponents?

The nth root of a is equivalent to a^(1/n). This means √a = a^(1/2) and ∛a = a^(1/3).

How do I simplify a root?

Factor the radicand into perfect powers and remaining factors. For example, √72 = √(36 × 2) = 6√2, since 36 is a perfect square.

What are perfect roots?

Perfect roots are numbers whose root is an integer. Examples: √25 = 5, ∛64 = 4, ∜16 = 2.