A surd is a root that cannot be simplified to a whole number, such as √2 or √3. Leaving answers as surds keeps them exact. To simplify a surd, take out the largest square factor: √75 = √25 × √3 = 5√3. Like surds can be added and subtracted, √a × √b = √(ab), and a surd is removed from a denominator by rationalising.
Surd rules
- √(ab) = √a × √b
- √(a/b) = √a / √b
- √a × √a = a
- Only like surds can be added: 5√2 + 3√2 = 8√2
- Rationalise: a/√b = a√b / b
What is a surd?
A surd is an irrational root, so its decimal never ends or repeats: √2 = 1.41421… Leaving the answer as √2 keeps it exact, which is what exam questions mean by “give your answer in surd form” or “exact form”. √9 is not a surd, because it equals 3.
Simplifying surds
Split the number under the root into a product that includes the largest possible square number, then take the square root of that factor. √75 = √(25 × 3) = 5√3. Using a smaller square factor, such as √48 = √(4 × 12) = 2√12, leaves more simplifying to do.
Adding, subtracting and multiplying surds
You can only add or subtract like surds, those with the same number under the root, in the same way as collecting like terms. Simplify each surd first: √18 + √2 = 3√2 + √2 = 4√2. To multiply, use √a × √b = √(ab), and remember that √a × √a = a.
Expanding brackets with surds
Expand surd brackets exactly as you would algebraic brackets, then simplify. In (2 + √5)(3 − √5), the four products are 6, −2√5, 3√5 and −5, which collect to 1 + √5. Squaring a bracket gives three terms: (3 + √2)² = 9 + 6√2 + 2.
Rationalising the denominator
It is conventional not to leave a surd in the denominator. Multiply the top and bottom by that surd: 12/√3 = 12√3/3 = 4√3. At A-Level you will also rationalise denominators such as 2 + √3, using the difference of two squares.
Worked examples
Example 1. Simplify √75.
- The largest square factor of 75 is 25.
- √75 = √25 × √3.
Answer: 5√3.
Example 2. Simplify √18 + √2.
- √18 = √9 × √2 = 3√2.
- 3√2 + √2.
Answer: 4√2.
Example 3. Work out √27 × √3.
- √27 × √3 = √81.
Answer: 9.
Example 4. Expand and simplify (2 + √5)(3 − √5).
- 6 − 2√5 + 3√5 − 5.
- Collect: (6 − 5) + (−2 + 3)√5.
Answer: 1 + √5.
Example 5. Rationalise the denominator of 12/√3.
- Multiply top and bottom by √3: 12√3/3.
- Simplify: 12 ÷ 3 = 4.
Answer: 4√3.
Example 6. Expand (3 + √2)².
- (3 + √2)(3 + √2) = 9 + 3√2 + 3√2 + 2.
- Collect.
Answer: 11 + 6√2.
Common mistakes
- Adding unlike surds. √2 + √3 is not √5.
- Not using the largest square factor. √48 = 4√3, not 2√12.
- Squaring a bracket term by term. (3 + √2)² is not 9 + 2; there is a middle term 6√2.
- Leaving a surd in the denominator. Multiply the top and bottom by the surd to rationalise.
- Forgetting √a × √a = a. √5 × √5 = 5, not √25 left unsimplified.
How surds are examined
Surds are Higher tier content in Edexcel GCSE (1MA1) and Edexcel International GCSE (4MA1), tested on non-calculator papers. Questions ask you to:
- simplify a surd, or write it in the form a√b
- add, subtract or multiply surds
- expand brackets and give the answer in the form a + b√c
- rationalise a denominator
Practise using the GCSE number revision engine, and the powers and roots worksheet for the underlying skills.
Surds appear in Pythagoras and exact trigonometry answers. At A-Level, see indices and surds.
Frequently asked questions
What is a surd?
A root that cannot be written as a whole number or fraction, such as √2 or √7.
How do you simplify a surd?
Write the number under the root as a square number times another number, then take the square root of the square: √50 = 5√2.
How do you rationalise a denominator?
Multiply the top and bottom of the fraction by the surd in the denominator.