Free Surds Calculator with Steps
Simplify surds, rationalise denominators, add, multiply and divide with animated factor trees, interactive number lines, and step-by-step working. Perfect for GCSE and A-Level maths exams.
Enter a positive number to simplify (e.g., 72, 50, 200)
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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" — reflecting how these numbers cannot "speak" as exact fractions.
Formally, √a is a surd when a is a positive integer that is NOT a perfect square. This means √a is irrational — its decimal expansion goes on forever without repeating.
For example, √2, √3, √5, √7 are all surds because they cannot be written as exact fractions. However, √4 = 2, √9 = 3, and √25 = 5 are NOT surds — they simplify to whole numbers.
In GCSE and A-Level maths, examiners award marks for leaving answers in "exact form" using surds, rather than writing decimal approximations. Writing 6√2 is exact; writing 8.485... is not.
| Expression | Value | Surd? |
|---|---|---|
| √2 | 1.41421... | Yes |
| √4 | 2 | No |
| √7 | 2.64575... | Yes |
| √9 | 3 | No |
| 3√5 | 6.70820... | Yes |
| √16 | 4 | No |
Surd Laws & Rules
Product Rule
√(ab) = √a × √b
Split surds by multiplying radicands
Quotient Rule
√(a/b) = √a / √b
Divide surds by dividing radicands
Like Surds Addition
a√n + b√n = (a+b)√n
Combine coefficients of like surds
Squaring Rule
(√a)² = a
Squaring a surd removes the root
Conjugate Rule
(a+√b)(a−√b) = a²−b
Difference of squares eliminates surds
Rationalisation
Multiply by √n/√n or conjugate
Remove surds from the denominator
Types of Surds
| Type | Notation | Example | Can Combine? |
|---|---|---|---|
| Simple surd | √n | √2, √5, √13 | N/A |
| Like surds | a√n, b√n | 2√3 and 5√3 | Yes — add coefficients |
| Unlike surds | a√m, b√n | 2√3 and 4√5 | No — different radicands |
| Compound surd | a + b√n | 2 + 3√5 | Only the rational parts |
| Binomial surd | (a + b√n) | (1 + √2) | Expand using FOIL |
| Conjugate surds | (a+√b) and (a−√b) | (3+√2) and (3−√2) | Multiply → a²−b |
Perfect Squares Reference
Memorising these squares helps you quickly identify factors when simplifying surds. If the number under the √ is divisible by any of these, you can simplify.
Which Method Do I Use?
Is it a single √n?
→ Simplify
Find the largest perfect square factor and extract it.
Same number under the √?
→ Add/Subtract coefficients
Like surds: a√n ± b√n = (a±b)√n. Simplify first if needed.
Surd in the denominator?
→ Rationalise
Simple: multiply by √n/√n. Binomial: multiply by conjugate.
(a+b√c)(d+e√f)?
→ Expand with FOIL
First, Outside, Inside, Last. Then collect like surds.
Common Mistakes to Avoid
Adding Unlike Surds
√2 + √3 = √5
√2 + √3 (cannot simplify)
Only like surds combine. √2 + √3 stays as it is.
Addition Instead of Product Rule
√12 = √4 + √8
√12 = √4 × √3 = 2√3
Use √(ab) = √a × √b (multiplication), never addition.
Forgetting to Simplify First
"√8 + √2 can't add"
√8 = 2√2, so √8 + √2 = 3√2
Always simplify surds before trying to add or subtract.
Surd Left in Denominator
5/√2 as final answer
5/√2 = 5√2/2
Always rationalise — exams require rational denominators.
Wrong Conjugate Sign
Conjugate of (3+√2) is (3+√2)
Conjugate of (3+√2) is (3−√2)
The conjugate changes the sign between the terms.
Not Fully Simplifying
Leaving √12 as final answer
√12 = √(4×3) = 2√3
Check: does the radicand have perfect square factors?
Worked Examples
Simplify √200
Step 1: Find perfect square factors of 200
200 = 100 × 2 (100 is the largest)
Step 2: Apply √(ab) = √a × √b
√200 = √100 × √2
Step 3: Simplify √100 = 10
= 10 × √2
Answer: 10√2
Rationalise 6/√3
Step 1: Multiply top and bottom by √3
(6/√3) × (√3/√3)
Step 2: Numerator: 6 × √3 = 6√3
Step 3: Denominator: √3 × √3 = 3
= 6√3 / 3
Step 4: Simplify the fraction
Answer: 2√3
Rationalise 5/(2+√3)
Step 1: Identify conjugate of (2+√3) → (2−√3)
Step 2: Multiply top and bottom by (2−√3)
5(2−√3) / (2+√3)(2−√3)
Step 3: Expand numerator: 10 − 5√3
Step 4: Denominator: 2² − (√3)² = 4 − 3 = 1
Answer: 10 − 5√3
Expand (3+√2)(1−2√2)
Step 1: FOIL — First: 3 × 1 = 3
Step 2: Outside: 3 × (−2√2) = −6√2
Step 3: Inside: √2 × 1 = √2
Step 4: Last: √2 × (−2√2) = −2 × 2 = −4
= 3 − 6√2 + √2 − 4
Step 5: Collect like terms
Answer: −1 − 5√2
Surds vs Decimals — Why Use Exact Form?
Examiners award marks for exact (surd) form. Decimals are approximations that lose precision — surds capture the exact value.
| Surd (Exact) | Decimal (Approximate) | Exact? |
|---|---|---|
| √2 | 1.41421356... | No — infinite decimal |
| 3√5 | 6.70820393... | No — infinite decimal |
| 6√2 | 8.48528137... | No — infinite decimal |
| 2√3 | 3.46410162... | No — infinite decimal |
| 10√2 | 14.14213562... | No — infinite decimal |
Decimal values go on forever without repeating. Only surd form is exact.
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. The term "surd" comes from the Latin "surdus" meaning deaf/mute.
How do you simplify surds?
Find the largest perfect square factor, use √(ab) = √a × √b to split it, then simplify the perfect square part. For example, √72 = √(36×2) = 6√2.
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. Unlike surds (e.g. √2 + √3) cannot be combined.
Why rationalise the denominator?
Rationalising produces cleaner answers that are easier to compare and combine. In GCSE and A-Level exams, marks are awarded for rationalised answers — leaving a surd in the denominator loses marks.
What is a conjugate?
The conjugate of (a + √b) is (a - √b). Multiplying conjugates uses the difference of squares to eliminate surds: (a + √b)(a - √b) = a² - b.
Is this aligned with GCSE/A-Level?
Yes! It covers simplifying, adding, multiplying, rationalising (simple and binomial), and FOIL expansion as required by Edexcel, AQA, and OCR exam boards.
Can you add √2 and √3?
No. √2 and √3 are unlike surds — they have different radicands. The expression √2 + √3 is already in its simplest form. Writing √2 + √3 = √5 is a common mistake.
What is the difference between surds and radicals?
"Surds" (UK term) and "radicals" (US term) refer to the same concept — irrational roots like √2 and √3. The UK curriculum uses "surds" exclusively.
How do I know when a surd is fully simplified?
A surd √n is fully simplified when n has no perfect square factors other than 1. Test: is n divisible by 4, 9, 16, 25, or 36? If yes, simplify further.
Why does multiplying conjugates eliminate the surd?
It uses the algebraic identity (a+b)(a-b) = a²-b². When applied to surds: (p+√q)(p-√q) = p²-(√q)² = p²-q. Both terms are rational, so the surd vanishes.
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