GCSE Surds Practice Questions

Free GCSE Surds practice questions with full step-by-step worked solutions. Covers simplifying surds, multiplying surds, adding surds, subtracting surds. Practise exam-style problems and check your method.

simplifying surdsmultiplying surdsadding surdssubtracting surdsdividing surdssquaring surds
GCSE Higher70 questionsStep-by-step solutions
Question 1
2 markseasy
Simplify 8\sqrt{8}.
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Worked solution

  1. Find the largest square factor of 8

    8=4×28 = 4 \times 2

    Look for the biggest perfect square dividing 8; here 4=224 = 2^2.

  2. Split the surd

    8=4×2\sqrt{8} = \sqrt{4} \times \sqrt{2}

    Use ab=a×b\sqrt{ab} = \sqrt{a}\times\sqrt{b} to separate the square part.

  3. Root the square part

    4=2\sqrt{4} = 2

    4\sqrt{4} is exactly 2.

  4. Write the simplified surd

    222\sqrt{2}

    So 8=22\sqrt{8} = 2\sqrt{2}.

Answer
222\sqrt{2}
Question 2
2 markseasy
Simplify 75\sqrt{75}.
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Worked solution

  1. Find the largest square factor of 75

    75=25×375 = 25 \times 3

    Look for the biggest perfect square dividing 75; here 25=5225 = 5^2.

  2. Split the surd

    75=25×3\sqrt{75} = \sqrt{25} \times \sqrt{3}

    Use ab=a×b\sqrt{ab} = \sqrt{a}\times\sqrt{b} to separate the square part.

  3. Root the square part

    25=5\sqrt{25} = 5

    25\sqrt{25} is exactly 5.

  4. Write the simplified surd

    535\sqrt{3}

    So 75=53\sqrt{75} = 5\sqrt{3}.

Answer
535\sqrt{3}
Question 3
3 marksintermediate
Rationalise the denominator of 82\dfrac{8}{\sqrt{2}}.
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Worked solution

  1. Multiply top and bottom by 2\sqrt{2}

    82×22\dfrac{8}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}}

    Clear the surd from the bottom.

  2. Multiply the numerators

    8×2=828 \times \sqrt{2} = 8\sqrt{2}

    The top becomes 828\sqrt{2}.

  3. Multiply the denominators

    2×2=2\sqrt{2}\times\sqrt{2} = 2

    The bottom becomes 2.

  4. Write the fraction

    822\dfrac{8\sqrt{2}}{2}

    Now cancel the numbers.

  5. Cancel

    82=4\dfrac{8}{2} = 4

    8 divided by 2 is 4.

  6. State the answer

    424\sqrt{2}

    So 8/2=428/\sqrt{2} = 4\sqrt{2}.

Answer
424\sqrt{2}
Question 4
4 markshard
Expand and simplify (3+2)(3+4)\left(\sqrt{3} + 2\right)\left(\sqrt{3} + 4\right).
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Worked solution

  1. Set up the expansion (FOIL)

    3⋅3+3⋅4+2⋅3+2⋅4\sqrt{3}\cdot\sqrt{3} + \sqrt{3}\cdot 4 + 2\cdot\sqrt{3} + 2\cdot 4

    Multiply every term in the first bracket by every term in the second.

  2. First product

    3×3=3\sqrt{3}\times\sqrt{3} = 3

    A surd times itself gives 3.

  3. Outer product

    3×4=43\sqrt{3}\times 4 = 4\sqrt{3}

    Coefficient in front of the surd.

  4. Inner product

    2×3=232\times\sqrt{3} = 2\sqrt{3}

    Another surd term.

  5. Last product

    2×4=82\times 4 = 8

    A whole number.

  6. Write all four terms

    3+43+23+83 + 4\sqrt{3} + 2\sqrt{3} + 8

    Line up the four products.

  7. Collect the whole numbers

    3+8=113 + 8 = 11

    Add the rational parts.

  8. Collect the surds

    43+23=634\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}

    Add the coefficients of 3\sqrt{3}.

  9. State the answer

    11+6311 + 6\sqrt{3}

    So the expansion is 11+6311 + 6\sqrt{3}.

  10. Decimal check

    ≈21.392\approx 21.392

    Both the product and the answer equal about 21.392.

Answer
11+6311 + 6\sqrt{3}
Question 5
4 markschallenging
Ravi says (2+3)(2−3)\left(2 + \sqrt{3}\right)\left(2 - \sqrt{3}\right) is irrational. Is he right? Explain.
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Worked solution

  1. Recognise difference of two squares

    (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2

    Here a=2a = 2 and b=3b = \sqrt{3}.

  2. Square the first

    22=42^2 = 4

    a2=4a^2 = 4.

  3. Square the second

    (3)2=3\left(\sqrt{3}\right)^2 = 3

    b2=3b^2 = 3.

  4. Subtract

    4−3=14 - 3 = 1

    So the product is 1.

  5. Check by full expansion

    4−23+23−34 - 2\sqrt{3} + 2\sqrt{3} - 3

    The 3\sqrt{3} terms cancel.

  6. Confirm cancellation

    −23+23=0-2\sqrt{3} + 2\sqrt{3} = 0

    Leaving 4−3=14 - 3 = 1.

  7. Judge the claim

    1 is rational1 \text{ is rational}

    1 is a whole number, so it is rational.

  8. Conclude Ravi is wrong

    Ravi is incorrect\text{Ravi is incorrect}

    The product is 1, which is rational, so Ravi is wrong.

  9. Reason

    conjugate surds\text{conjugate surds}

    Conjugate surds always multiply to a rational number.

  10. Confirm

    11

    The exact value is 1.

  11. Underline the final answer

    = No - it equals 1, which is rational.

    Write the answer on its own line so it is clear and easy to mark.

  12. Check it answers the question

    No - it equals 1, which is rational.

    Re-reading the question, this is exactly what was required.

  13. State the conclusion in one line

    No - it equals 1, which is rational.

    The working above leads directly to this result.

  14. Match to the options

    No - it equals 1, which is rational.

    Compare each choice; only the option agreeing with this working is correct.

  15. Sense check

    No - it equals 1, which is rational.

    The result is consistent with the decimal estimates found along the way.

Answer
No - it equals 1, which is rational.

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