Surds Calculator

Simplify surds and radical expressions. Input a number or expression. Get the simplified surd, decimal approximation, and step-by-step solution. A great tool for students. Informational only—consult a teacher.

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Published: October 23, 2025 | Updated: October 23, 2025 | Reviewed by: Science/Math Editor

Input Expression

Enter a surd expression. Use √ for square root.

Preset Surds

Surd Rules Reference

  • √(a×b) = √a × √b
  • √(a÷b) = √a ÷ √b
  • √(a²) = a (for positive a)
  • (√a)² = a

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How it works

Simplified Surd = Coefficient × √(Extracted Squares) × √(Remaining Radicand). We find the prime factors of the radicand, extract perfect squares, and simplify the expression.

For example, to simplify 2√18:

  1. Find prime factors of 18: 18 = 2 × 3 × 3
  2. Identify perfect squares: 3² = 9
  3. Extract: 2√18 = 2√(9×2) = 2 × 3√2 = 6√2
  4. Decimal approximation: 6 × 1.4142 ≈ 8.4853

Inputs explained

  • Input Expression: Enter a surd expression using the √ symbol. Examples: "√18", "2√18", "3√12".
  • Coefficient: The number outside the radical (optional, defaults to 1).
  • Radicand: The number under the square root symbol.
  • Decimal Precision: Number of decimal places for the approximation (0-10).

Example

Input: 2√18

Step 1: Prime factorization of 18 = 2 × 3²

Step 2: Extract perfect square: √(3²) = 3

Step 3: Simplify: 2√18 = 2 × 3 × √2 = 6√2

Decimal: 6 × 1.4142 ≈ 8.4853

Tips & notes

  • A surd is a radical expression, like √2, that cannot be simplified to a rational number.
  • To simplify, look for perfect square factors inside the radical (4, 9, 16, 25, 36, etc.).
  • The decimal approximation is useful for practical calculations and comparisons.
  • Not all square roots are surds—√4 = 2 is rational, so it's not technically a surd.
  • When multiplying surds with the same radicand, multiply the coefficients: 2√3 × 5√3 = 10 × 3 = 30.

FAQs

A mathematical term for a root, like a square root. For example, √2, √3, and √5 are all surds. These are irrational numbers that cannot be expressed as exact fractions or finite decimals.

Factor the radicand (number under the root), extract perfect squares, and simplify. For example, √18 = √(9×2) = 3√2. This works because √9 = 3.

Approximately 1.4142. The exact value is irrational and cannot be expressed as a finite decimal. It continues infinitely without repeating.

Only like surds (same radicand) can be added. For example, 2√3 + 5√3 = 7√3, but √2 + √3 cannot be simplified further.

Multiply the coefficients and the radicands separately. For example, 2√3 × 5√2 = (2×5)√(3×2) = 10√6.

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Disclaimer

Informational mathematical tool for learning and practice. Consult a teacher for educational guidance.

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