Simplify square roots exactly, expand brackets containing surds and rationalise denominators without a calculator. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A surd is a root that will not simplify to an integer: 12=23 because 12=4×3. Simplify by taking out square factors. Add and subtract like surds only: 53+23=73, but 3+2 stays as it is. Multiply: ab=ab. Rationalise a denominator by multiplying top and bottom by the surd (or by the conjugate c−de if there are two terms).
Exact trig and Pythagoras answers want simplified surds, not decimals. 36=363=23. Expand (2+5)2=4+45+5=9+45. Check by squaring back when you have simplified a single root.
Pull out square factors: 48=16×3=43. Rationalise ba by multiplying top and bottom by b. Expand (p+q)2 with four products.
Objectives
Take square factors out of a root.
Add and subtract surds that share a root.
Rationalise a denominator, including one with a bracket like 3 + √2, and simplify.
Key points
Simplify a surd by pulling out the largest square factor: 72=62.
Add and subtract like surds: 32+52=82.
Expand brackets with surds, including squaring a bracket.
Rationalise a denominator of the form a or b±a.
Worked examples
Example 1
Simplify the surd √18.
2 marks
Find the largest square factor of 18, which is 9
Write √18 = √(9×2) = √9 × √2 = 3√2
The simplified surd is 3√2
Answer: 3√2
Common slips
Slip: Use 6 and 3 as factors of 18 Neither 6 nor 3 is a square number, so the root does not simplify.
Slip: Write √18 = 9√2 9 is the square factor inside the root, so its square root 3 comes outside.
Slip: Leave the answer as √(9×2) Evaluate the square root of the square factor.
Example 2
Add the surds √32 + √18.
2 marks
Simplify each surd first: √32 = 4√2 and √18 = 3√2
Combine the like terms: 4√2 + 3√2 = (4+3)√2 = 7√2
The simplified sum is 7√2
Answer: 7√2
Common slips
Slip: Add the numbers under the roots to get √50 You cannot add the numbers inside roots unless they share the same root.
Slip: Multiply 4√2 by 3√2 to get 24 This is addition of like surds, not multiplication.
Slip: Write 7√4 The common root √2 remains unchanged when adding coefficients.
Example 3
Rationalise and simplify 3/√12.
3 marks
Simplify √12 first as 2√3, giving 3/(2√3)
Multiply numerator and denominator by √3 to get 3√3 / 6
Cancel the factor of 3 to leave √3/2
Answer: √3/2
Common slips
Slip: Multiply numerator and denominator by 12 Simplifying the denominator first keeps the numbers smaller.
Slip: Cancel 3 with 12 inside the root You cannot cancel a number outside a root with a number inside a root.
Slip: Write 3√3/6 and stop Always reduce numerical fractions to lowest terms.
Questions and answers
GCSE Maths surds questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Simplify the surd √48.
Show answer
Answer: 4√3
Find the largest square factor: 48 = 16 × 3.
Take √16 = 4 outside the root to get 4√3.
M1 — Find the largest square factor: 48 = 16 × 3.
A1 — Take √16 = 4 outside the root to get 4√3.
Question 2 Easy2 marks
Simplify the surd √50.
Show answer
Answer: 5√2
Find the largest square factor: 50 = 25 × 2.
Take √25 = 5 outside the root to get 5√2.
M1 — Find the largest square factor: 50 = 25 × 2.
A1 — Take √25 = 5 outside the root to get 5√2.
Question 3 Medium2 marks
Add the surds √8 + √18.
Show answer
Answer: 5√2
Simplify each root: √8 = 2√2 and √18 = 3√2.
Add the coefficients: 2√2 + 3√2 = 5√2.
M1 — Simplify each root: √8 = 2√2 and √18 = 3√2.
A1 — Add the coefficients: 2√2 + 3√2 = 5√2.
Question 4 Medium2 marks
Add the surds √12 + √27.
Show answer
Answer: 5√3
Simplify each root: √12 = 2√3 and √27 = 3√3.
Add the coefficients: 2√3 + 3√3 = 5√3.
M1 — Simplify each root: √12 = 2√3 and √27 = 3√3.
A1 — Add the coefficients: 2√3 + 3√3 = 5√3.
Question 5 Hard2 marks
Rationalise the denominator of 5/√5.
Show answer
Answer: √5
Multiply top and bottom by √5 to get 5√5/5.
Cancel the common factor of 5 to leave √5.
M1 — Multiply top and bottom by √5 to get 5√5/5.
A1 — Cancel the common factor of 5 to leave √5.
Question 6 Hard2 marks
Rationalise the denominator of 6/√3.
Show answer
Answer: 2√3
Multiply top and bottom by √3 to get 6√3/3.
Divide 6 by 3 to get 2√3.
M1 — Multiply top and bottom by √3 to get 6√3/3.
A1 — Divide 6 by 3 to get 2√3.
Question 7 Exam-style4 marks
Write (√2 + 3)² − 6/√2 in the form a + b√2, where a and b are integers.