Free Surds Calculator with Steps
Simplify surds, rationalise denominators, add, multiply and divide with step-by-step working. Perfect for GCSE and A-Level maths exams.
Enter a positive number to simplify (e.g., 72, 50, 200)
Key Surd Rules
√a × √b = √(ab)
Multiply surds by multiplying the radicands.
√a ÷ √b = √(a/b)
Divide surds by dividing the radicands.
√(a²b) = a√b
Extract perfect square factors from under the root.
a/√b = a√b/b
Multiply top and bottom by √b.
Common Mistakes
Adding unlike surds
Only like surds (same number under √) can be combined: 2√3 + 5√3 = 7√3
Forgetting to simplify first
Always simplify surds before adding/subtracting to find like terms.
What is a Surd?
A surd is an irrational number expressed as a root that cannot be simplified to remove the root sign. The word comes from the Latin "surdus" meaning "deaf" or "mute".
For example, √2, √3, √5 are surds because they cannot be written as exact fractions. However, √4 = 2 and √9 = 3 are NOT surds because they simplify to whole numbers.
In GCSE and A-Level maths, you'll often need to leave answers in "exact form" using surds, rather than writing decimal approximations.
The Key Rule
√(a × b) = √a × √b
Key Operations
Simplifying Surds
Find the largest perfect square factor
√72 = √(36 × 2) = 6√2
√50 = √(25 × 2) = 5√2
Adding Like Surds
Only surds with the same radicand can be combined
3√2 + 5√2 = 8√2 ✓
√2 + √3 = cannot simplify ✗
Rationalising Denominators
Remove surds from the denominator
Simple:
1/√2 → √2/2
Binomial:
1/(1+√2)
💡 Exam Tip
Always simplify surds BEFORE adding or subtracting! √8 + √2 looks impossible, but √8 = 2√2, so √8 + √2 = 2√2 + √2 = 3√2.
Common Mistakes to Avoid
❌ Adding Unlike Surds
√2 + √3 = √5
√2 + √3 cannot be simplified. Only like surds combine.
❌ Wrong Split
√12 = √4 + √8
Use √(ab) = √a × √b, not addition. √12 = √4 × √3 = 2√3
❌ Forgetting to Simplify
"√8 + √2 can't add"
Always simplify first! √8 = 2√2, so √8 + √2 = 3√2
❌ Surd in Denominator
5/√2 as final answer
Always rationalise! 5/√2 = 5√2/2
Worked Examples
Example 1: Simplify √72
Step 1: Find perfect square factors of 72
72 = 36 × 2 (36 is the largest)
Step 2: Apply √(ab) = √a × √b
√72 = √36 × √2
Step 3: Simplify √36
= 6 × √2
Answer: 6√2
Example 2: Rationalise 6/√3
Step 1: Multiply top and bottom by √3
6/√3 × √3/√3
Step 2: Simplify numerator
= 6√3 / (√3 × √3)
Step 3: Simplify denominator
= 6√3 / 3
Answer: 2√3
Frequently Asked Questions
What is a surd?
A surd is an irrational number expressed as a root that cannot be simplified. √2 is a surd, but √4 = 2 is not.
How do you simplify surds?
Find the largest perfect square factor, use √(ab) = √a × √b to split it, then simplify the perfect square part.
What are like surds?
Like surds have the same radicand (number under the root). You can only add/subtract like surds: 2√3 + 5√3 = 7√3.
Why rationalise the denominator?
Rationalising makes answers cleaner and easier to compare. In exams, marks are often given for rationalised answers.
What is a conjugate?
The conjugate of (a + √b) is (a - √b). Multiplying conjugates eliminates surds: (a + √b)(a - √b) = a² - b.
Is this aligned with GCSE/A-Level?
Yes! It covers simplifying, adding, multiplying, and rationalising as required by Edexcel, AQA, and OCR.
Explore More Free Tools
All our tools are 100% free with step-by-step learning