Surd Simplifier

What is a Surd?

A surd is a square root (or other root) that cannot be simplified to a rational number. For example, √2, √3, and √5 are surds because they cannot be expressed as exact fractions. √4 is not a surd because it simplifies to 2.

How to Simplify Surds

To simplify a surd, find the largest perfect square that divides the number under the root sign. Then use the property √(a × b) = √a × √b.

• Step 1: Find the largest perfect square factor of the number under the root
• Step 2: Split the surd: √(a × b) = √a × √b
• Step 3: Simplify √a to a whole number (since a is a perfect square)
• Step 4: The simplified form is a√b where b has no square factors

Perfect Squares to Know

• 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
• 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100
• 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
• 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400

Common Surd Simplifications

• √8 = 2√2
• √12 = 2√3
• √18 = 3√2
• √20 = 2√5
• √27 = 3√3
• √32 = 4√2
• √45 = 3√5
• √48 = 4√3
• √50 = 5√2
• √72 = 6√2
• √75 = 5√3
• √98 = 7√2

Why Use Our Surd Simplifier?

• ✅ 100% Free — No hidden fees or subscriptions
• ✅ No Registration — Use instantly without creating an account
• ✅ Step-by-Step Working — Shows full simplification process
• ✅ Identifies perfect square factors automatically
• ✅ Decimal approximation for reference
• ✅ Mobile friendly — Works perfectly on any device
• ✅ UK Curriculum aligned — Perfect for GCSE and A-Level Maths

Frequently Asked Questions

• What is the difference between a surd and a radical? — A surd is a radical that cannot be simplified to a rational number. All surds are radicals, but not all radicals are surds.
• How do I simplify √18? — √18 = √(9 × 2) = √9 × √2 = 3√2.
• Can I simplify surds with coefficients? — Yes! For example, 3√50 = 3 × √(25 × 2) = 3 × 5√2 = 15√2.
• What if the number under the root is prime? — If the number is prime (like 2, 3, 5, 7, 11), it cannot be simplified further.
• What is a surd in GCSE Maths? — Surds are irrational square roots that appear in the GCSE and A-Level Maths curriculum, often in questions involving simplification, rationalisation, and algebraic manipulation.

Advanced Surd Operations

Once simplified, surds can be added, subtracted, multiplied, and divided:

• Addition: a√b + c√b = (a + c)√b (only like surds can be added)
• Multiplication: √a × √b = √(a × b)
• Rationalising the denominator: To remove surds from a denominator, multiply numerator and denominator by the conjugate or the surd itself.