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Surds practice questions with answers

Higher · Grade 7 · Non-calculator

A second set of surds questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

Simplify the surd √12.

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Answer: 2√3

  1. Find the largest square factor: 12 = 4 × 3.
  2. Take √4 = 2 outside the root to get 2√3.
  • Method (1): Find the largest square factor: 12 = 4 × 3.
  • Accuracy (1): Take √4 = 2 outside the root to get 2√3.

Question 2 Easy

Simplify the surd √72.

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Answer: 6√2

  1. Find the largest square factor: 72 = 36 × 2.
  2. Take √36 = 6 outside the root to get 6√2.
  • Method (1): Find the largest square factor: 72 = 36 × 2.
  • Accuracy (1): Take √36 = 6 outside the root to get 6√2.

Question 3 Medium

Subtract the surds √50 - √8.

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Answer: 3√2

  1. Simplify each root: √50 = 5√2 and √8 = 2√2.
  2. Subtract the coefficients: 5√2 - 2√2 = 3√2.
  • Method (1): Simplify each root: √50 = 5√2 and √8 = 2√2.
  • Accuracy (1): Subtract the coefficients: 5√2 - 2√2 = 3√2.

Question 4 Medium

Rationalise the denominator of 1/√3.

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Answer: √3/3

  1. Multiply numerator and denominator by √3.
  2. 1 × √3 / (√3 × √3) = √3/3.
  • Method (1): Multiply numerator and denominator by √3.
  • Accuracy (1): 1 × √3 / (√3 × √3) = √3/3.

Question 5 Hard

Rationalise the denominator of 1/(3 + √2).

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Answer: (3 − √2)/7

  1. Multiply top and bottom by the conjugate 3 − √2.
  2. Bottom: (3 + √2)(3 − √2) = 9 − 2 = 7. Top: 3 − √2.
  • Method (1): Multiply top and bottom by the conjugate 3 − √2.
  • Accuracy (1): Bottom: (3 + √2)(3 − √2) = 9 − 2 = 7. Top: 3 − √2.

Question 6 Hard

Expand and simplify (√3 + 1)².

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Answer: 4 + 2√3

  1. Expand: (√3)² + 2(√3)(1) + 1² = 3 + 2√3 + 1.
  2. Collect integer terms: 3 + 1 = 4, giving 4 + 2√3.
  • Method (1): Expand: (√3)² + 2(√3)(1) + 1² = 3 + 2√3 + 1.
  • Accuracy (1): Collect integer terms: 3 + 1 = 4, giving 4 + 2√3.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Rationalise the denominator and simplify 8/√2.

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Answer: 4√2

  1. Multiply top and bottom by √2.
  2. 8 × √2 / (√2 × √2) = 8√2 / 2.
  3. Cancel 8/2 = 4 to leave 4√2.
  • Method (1): Multiply numerator and denominator by √2, giving 8√2 / 2
  • Accuracy (1): 4√2

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

Why does √48 simplify to 4√3?

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16 is a square factor of 48, and √16 is 4. 16 is a square factor of 48, and √16 is 4.

Check 2

What is the square factor inside √50?

  1. 25
  2. 50
  3. 26
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25 25 is the square factor, so √50 = 5√2.

Check 3

How do you add √8 and √18?

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Write 2√2 + 3√2, which gives 5√2. Write 2√2 + 3√2, which gives 5√2.

Check 4

Why can √12 and √27 be added?

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Both simplify to multiples of √3: 2√3 + 3√3 = 5√3. Both simplify to multiples of √3: 2√3 + 3√3 = 5√3.

Check 5

What is √50 minus √8?

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5√2 - 2√2 = 3√2. 5√2 - 2√2 = 3√2.

Check 6

What does it mean to rationalise a denominator?

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Rewrite the fraction so the bottom is not a surd. Rewrite the fraction so the bottom is not a surd.

Check 7

How do you rationalise 1/√3?

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Multiply top and bottom by √3 to get √3/3. Multiply top and bottom by √3 to get √3/3.

Check 8

Why does 5/√5 become √5?

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5√5/5 cancels to leave √5. 5√5/5 cancels to leave √5.

Check 9

Why does 6/√3 become 2√3?

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6√3/3 cancels by 3 to leave 2√3. 6√3/3 cancels by 3 to leave 2√3.

Check 10

What is the conjugate of 3 + √2?

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Change the sign between the terms. (3 + √2)(3 − √2) = 7. Change the sign between the terms. (3 + √2)(3 − √2) = 7.

Check 11

What is the rationalised form of 3/√12?

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3/(2√3) = 3√3/6 = √3/2. 3/(2√3) = 3√3/6 = √3/2.

Check 12

Expand (√3 + 1)² and name both parts.

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The rational part is 4 and the surd part is 2√3. The rational part is 4 and the surd part is 2√3.

Check 13

Why not replace √3 with 1.73 in an exact answer?

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The decimal is rounded, whereas the surd is exact. The decimal is rounded, whereas the surd is exact.

Check 14

What is √12 as a multiple of √3?

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√12 = √(4×3) = 2√3. √12 = √(4×3) = 2√3.

Check 15

Which surd added to √8 gives 5√2?

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√18, because 3√2 is the missing part. √18, because 3√2 is the missing part.

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