Surds
Surds are exact square root expressions that cannot be simplified to whole numbers. GCSE Higher students often meet surds in exact form questions.
√4 = 2, so this is not a surd.
√2 cannot be written exactly as a whole number, so √2 is a surd.
Simplifying surds
To simplify a surd, look for a square number factor inside the root.
Example 1: Simplify √20
Find a square factor of 20:
20 = 4 × 5
Split the root:
√20 = √4 × √5
Simplify √4:
√20 = 2√5
Example 2: Simplify √72
Use the square factor 36:
72 = 36 × 2
√72 = √36 × √2 = 6√2
Multiplying surds
When multiplying surds, multiply the numbers inside the roots.
Example 3: Multiply surds
Simplify √3 × √12.
√3 × √12 = √36
√36 = 6
Example 4: Multiply with coefficients
Simplify 2√5 × 3√10.
Multiply the coefficients and the surds:
2√5 × 3√10 = 6√50
Simplify √50:
6√50 = 6 × 5√2 = 30√2
Adding and subtracting surds
You can only add or subtract like surds. They must have the same root part.
Example 5: Add like surds
Simplify 3√2 + 5√2.
3√2 + 5√2 = 8√2
Example 6: Simplify before adding
Simplify √18 + √8.
√18 = 3√2
√8 = 2√2
3√2 + 2√2 = 5√2
Expanding brackets with surds
Expand brackets with surds using the same method as normal algebra.
Example 7: Expand and simplify
Expand 2√3(√3 + 4).
2√3 × √3 = 2√9 = 6
2√3 × 4 = 8√3
Answer: 6 + 8√3
Rationalising the denominator
Rationalising the denominator means removing a surd from the bottom of a fraction.
Example 8: Rationalise a simple denominator
Rationalise 5/√3.
Multiply top and bottom by √3:
5/√3 × √3/√3 = 5√3/3
A common mistake is writing √20 = √4 + √5. This is incorrect. Instead, √20 = √4 × √5 = 2√5.
In Higher GCSE questions, leave surd answers in exact simplified form unless the question asks for a decimal approximation.
Video explanation
A short Worthing Maths Tutor video explanation for surds can be embedded here later to improve student engagement and time on page.
Practice questions
- Simplify √28.
- Simplify √45.
- Simplify √2 × √18.
- Simplify 4√3 + 7√3.
- Rationalise 2/√5.
Answers
- 2√7
- 3√5
- 6
- 11√3
- 2√5/5
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