Simplifying Surds Questions with Answers

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Simplifying Surds Questions with Answers

Here are 14 simplifying surds questions with answers, split into 7 levels that start with √72 and finish with mixed questions that multiply, divide, add and subtract surds. For a fresh set of 10 questions at any level, open the same skill in our free Skills Lab, which also steps through the working one line at a time.

  • 7levels, easiest first
  • 14questions on this page, with answers
  • 10fresh questions every time in the Lab
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A worksheet shows the answer. The Skills Lab shows how to get there. Pick a level and the working appears one step at a time, each time you press the button, so your child can follow every move and then try ten questions of their own.

Skills Lab teach mode working through simplifying surds step by step
Teach: the working, one step at a timeThe square numbers light up beside the working, and the largest one that divides the number is the one taken out.
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How to simplify a surd

Find the largest square number that is a factor of the number under the root. Split the root into that square and what is left, then take the square root of the square out in front. When you add or subtract, simplify every surd first, then collect like surds the same way you collect like terms such as 2x + 3x.

Simplify √72

  1. The square numbers that divide 72 are 4, 9 and 36. The largest is 36.
  2. √72 = √36 × √2
  3. √36 = 6, so √72 = 6√2

Simplify 3√8

  1. The largest square number that divides 8 is 4.
  2. 3√8 = 3 × √4 × √2 = 3 × 2√2
  3. 3√8 = 6√2

Simplifying Surds Questions by level

Click a level to practise it
Level Question Question
Level 1Simplify a surd 1. √147 2. √252
Level 2A coefficient in front 3. 6√63 4. 3√126
Level 3Multiply surds 5. 5√13 × 5√5 6. 2√3 × 2√2
Level 4Divide surds 7. √255 ÷ √15 8. 12√30 ÷ 4√3
Level 5Add and subtract like surds 9. 4√3 − 7√3 10. 5√7 − 12√7 − 10√7
Level 6Simplify, then add or subtract 11. −√175 − √112 − √63 12. √125 − 2√5
Level 7Everything mixed 13. 2√14 − 9√14 14. −(3√10)²
Show all 14 answers
Question Answer
1. √147 7√3
2. √252 6√7
3. 6√63 18√7
4. 3√126 9√14
5. 5√13 × 5√5 25√65
6. 2√3 × 2√2 4√6
7. √255 ÷ √15 √17
8. 12√30 ÷ 4√3 3√10
9. 4√3 − 7√3 −3√3
10. 5√7 − 12√7 − 10√7 −17√7
11. −√175 − √112 − √63 −12√7
12. √125 − 2√5 3√5
13. 2√14 − 9√14 −7√14
14. −(3√10)² −90

Common mistakes

  • Taking out a square that is not the largest. √72 = 2√18 is true, but 18 still has 9 as a factor, so it is not fully simplified. Always check the number left under the root.
  • Adding the numbers under the roots. √2 + √3 is not √5. Only like surds can be collected: 2√3 + 5√3 = 7√3.
  • Forgetting the number already in front. In 3√8, the 2 that comes out multiplies the 3, giving 6√2, not 3 + 2√2.

Questions parents and students ask

What does it mean to simplify a surd?

It means writing the surd with the smallest possible whole number under the root, for example √72 = 6√2. You do it by taking out the largest square factor.

Is simplifying surds on GCSE Foundation or Higher?

Surds are on the GCSE Higher tier. Foundation students meet square roots but are not asked to simplify surds.

How do you add surds?

Simplify each surd first. If they then have the same number under the root, add the numbers in front: √12 + √27 = 2√3 + 3√3 = 5√3.

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