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.

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Put these formulas into practice with our instant, step-by-step Root Calculator.

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Use our free **Root Calculator** to **calculate any root of a number** instantly. This **nth root calculator** helps you **find square roots**, **cube roots**, and **higher-order roots** with simplified form. Whether you need to **simplify radicals** or **compute decimal approximations**, this tool provides accurate results. Features include **radical simplification**, **principal root identification**, and **step-by-step derivation**. 100% free — no signup required!

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)** into the root calculator.
  2. Step 2: Enter the **root index** (2 for square root, 3 for cube root, etc.).
  3. Step 3: Click Calculate to see the **exact root** and decimal approximation.
  4. Step 4: Review the **simplified radical form** of the result.
  5. Step 5: Check the **decimal approximation** for non-perfect roots.
  6. Step 6: Follow the **step-by-step derivation** showing how the root is computed.
  7. Step 7: Use the detailed breakdown to understand the **relationship between roots and exponents**.

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.

How do I simplify a radical expression?

Factor the radicand into perfect powers and remaining factors. For example, √72 = √(36 × 2) = 6√2, since 36 is a perfect square. Extract as many perfect powers as possible from under the radical.

What is the principal nth root?

The principal nth root is the real root when it exists. For even indices, it is the non-negative root. For odd indices, it has the same sign as the radicand.