Show WorkingWorked example 1: simplify, add and multiply surds
Simplify √50.
- Find the biggest square number that goes into 50. It is 25, because 50 = 25 × 2.
- √50 = √25 × √2 = 5√2.
Answer 5√2
NumberHigher
Surds GCSE questions with answers
8 surds questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.
Surds are roots that do not work out to a whole number or fraction, like √2. On Higher you simplify surds, add and multiply them, expand brackets with them and rationalise the denominator so there is no root on the bottom of a fraction.
01 / Quiz yourselfTry it, then tap to check
Surds questions to try
Work each one out on paper first. Then open the answer and compare your working with ours, step by step.
01 / Simplify, add and multiply surdsHigher only
Simplify √150.
Show the answer and working
- Find the biggest square number that goes into 150. It is 25, because 150 = 25 × 6.
- √150 = √25 × √6 = 5√6.
Answer 5√6
02 / Simplify, add and multiply surdsHigher only
Simplify √45 + √80. Give your answer as simply as you can.
Show the answer and working
- Simplify each surd so they all have √5 in them: √45 = √9 × 5 = 3√5 and √80 = √16 × 5 = 4√5.
- Now they are like terms, so collect them, just like 3x + 4x: 3√5 + 4√5 = 7√5.
Answer 7√5
03 / Simplify, add and multiply surdsHigher only
Simplify √300.
Show the answer and working
- Find the biggest square number that goes into 300. It is 100, because 300 = 100 × 3.
- √300 = √100 × √3 = 10√3.
Answer 10√3
04 / Simplify, add and multiply surdsHigher only
Simplify √112 − √7. Give your answer as simply as you can.
Show the answer and working
- Simplify each surd so they all have √7 in them: √112 = √16 × 7 = 4√7.
- Now they are like terms, so collect them, just like 4x − x: 4√7 − √7 = 3√7.
Answer 3√7
05 / Rationalise the denominator and expand surdsHigher only
Expand and simplify (5 + √2)(6 + √2).
Show the answer and working
- Multiply each term in the first bracket by each term in the second bracket.
- 5 × 6 = 30, 5 × √2 = 5√2, √2 × 6 = 6√2, and √2 × √2 = 2, because √2 × √2 = 2.
- Collect the whole numbers: 30 + 2 = 32. Collect the surds: 5√2 + 6√2 = 11√2.
- So the answer is 32 + 11√2.
Answer 32 + 11√2
06 / Rationalise the denominator and expand surdsHigher only
Rationalise the denominator of 10√10. Give your answer as simply as you can.
Show the answer and working
- Multiply the top and the bottom by √10. This does not change the value, because √10√10 = 1.
- Top: 10 × √10 = 10√10. Bottom: √10 × √10 = 10.
- So 10√10 = 10√1010, which simplifies to √10.
Answer √10
07 / Rationalise the denominator and expand surdsHigher only
Rationalise the denominator of 13√3. Give your answer as simply as you can.
Show the answer and working
- Multiply the top and the bottom by √3. This does not change the value, because √3√3 = 1.
- Top: 13 × √3 = 13√3. Bottom: √3 × √3 = 3.
- So 13√3 = 13√33.
Answer 13√33
08 / Rationalise the denominator and expand surdsHigher only
Expand and simplify (2 + 3√3)(2 − 3√3).
Show the answer and working
- Multiply each term in the first bracket by each term in the second bracket.
- 2 × 2 = 4, 2 × (−3√3) = −6√3, 3√3 × 2 = 6√3, and 3√3 × (−3√3) = −27, because √3 × √3 = 3.
- Collect the whole numbers: 4 − 27 = −23. The surds cancel out: −6√3 + 6√3 = 0.
- So the answer is −23.
Answer −23
02 / What's covered2 kinds of question
Surds at GCSE and where marks go
- Simplify, add and multiply surdsHigher only
- Rationalise the denominator and expand surdsHigher only
Worked example 2: rationalise the denominator and expand surds
Rationalise the denominator of 24√10. Give your answer as simply as you can.
- Multiply the top and the bottom by √10. This does not change the value, because √10√10 = 1.
- Top: 2 × √10 = 2√10. Bottom: 4√10 × √10 = 40, because √10 × √10 = 10.
- So 24√10 = 2√1040, which simplifies to √1020.
Answer √1020
A common mistake
Adding the numbers under the roots. √2 + √8 is not √10. Simplify first: √8 = √4 × √2 = 2√2, so √2 + √8 = 3√2.
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