Hard GCSE Surds Questions

Challenging, exam-style GCSE Surds questions with worked solutions. Stretch yourself on the hardest expanding surd brackets, difference of two squares, rationalising, conjugates problems.

expanding surd bracketsdifference of two squaresrationalisingconjugatesadding surdssimplifying surds
GCSE Higher34 questionsStep-by-step solutions
Question 1
4 markschallenging
Ravi says (2+3)(2−3)\left(2 + \sqrt{3}\right)\left(2 - \sqrt{3}\right) is irrational. Is he right? Explain.
Show worked solution

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.
Question 2
4 markschallenging
For positive whole numbers a and b, what does (a+b)(a−b)\left(\sqrt{a} + \sqrt{b}\right)\left(\sqrt{a} - \sqrt{b}\right) equal, and why is it always rational?
Show worked solution

Worked solution

  1. Recognise the structure

    (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2

    This is a difference of two squares with x=ax = \sqrt{a}, y=by = \sqrt{b}.

  2. Square the first

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

    Squaring a surd removes the root.

  3. Square the second

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

    Squaring removes the root.

  4. Subtract

    a−ba - b

    So the product is a - b.

  5. Why the surds vanish

    −ab+ab=0-\sqrt{ab} + \sqrt{ab} = 0

    The two cross terms are equal and opposite, so they cancel.

  6. Why it is rational

    a−b is a whole numbera - b \text{ is a whole number}

    Since a and b are whole numbers, a - b is rational.

  7. Test with numbers

    (5+2)(5−2)=5−2=3(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) = 5 - 2 = 3

    A concrete example gives 3, a whole number.

  8. Confirm the example

    ≈3\approx 3

    Numerically the product is exactly 3.

  9. Reason

    conjugate surds\text{conjugate surds}

    Multiplying conjugate surds always removes the roots.

  10. Conclude

    a−ba - b

    So the product is a - b, which is always rational.

  11. Underline the final answer

    =a−b= a - b

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

  12. Check it answers the question

    a−ba - b

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

  13. State the conclusion in one line

    a−ba - b

    The working above leads directly to this result.

  14. Match to the options

    a−ba - b

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

  15. Sense check

    a−ba - b

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

Answer
a−ba - b
Question 3
4 markschallenging
Why does 50\sqrt{50} simplify to 525\sqrt{2} rather than to some other surd?
Show worked solution

Worked solution

  1. Look for square factors of 50

    50=25×250 = 25 \times 2

    25 is a perfect square and 2 is left over.

  2. Check 25 is the largest

    50=2×25=5×1050 = 2\times25 = 5\times10

    Only 25 among the factors of 50 is a perfect square bigger than 1.

  3. Split the surd

    50=25×2\sqrt{50} = \sqrt{25}\times\sqrt{2}

    Use ab=a×b\sqrt{ab} = \sqrt{a}\times\sqrt{b}.

  4. Root the square part

    25=5\sqrt{25} = 5

    So 50=52\sqrt{50} = 5\sqrt{2}.

  5. Check the remainder

    2 is square-free2 \text{ is square-free}

    2\sqrt{2} cannot be reduced, so the surd is fully simplified.

  6. Why not 5105\sqrt{10}?

    (510)2=250≠50\left(5\sqrt{10}\right)^2 = 250 \neq 50

    5105\sqrt{10} squares to 250, so it is not equal to 50\sqrt{50}.

  7. Why not 2\sqrt{2}?

    (2)2=2≠50\left(\sqrt{2}\right)^2 = 2 \neq 50

    2\sqrt{2} alone is far too small.

  8. Decimal check

    52≈7.0715\sqrt{2}\approx 7.071

    50\sqrt{50} is also about 7.071.

  9. Reason

    take out the biggest square\text{take out the biggest square}

    Simplifying means removing the largest perfect-square factor.

  10. Conclude

    50=52\sqrt{50} = 5\sqrt{2}

    So the correct simplification is 525\sqrt{2}.

  11. Underline the final answer

    =52= 5\sqrt{2}

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

  12. Check it answers the question

    525\sqrt{2}

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

  13. State the conclusion in one line

    525\sqrt{2}

    The working above leads directly to this result.

  14. Match to the options

    525\sqrt{2}

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

  15. Sense check

    525\sqrt{2}

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

Answer
525\sqrt{2}
Question 4
4 markschallenging
What is 12\dfrac{1}{\sqrt{2}} with a rationalised denominator, and why?
Show worked solution

Worked solution

  1. State the goal

    remove the surd from the bottom\text{remove the surd from the bottom}

    A rationalised denominator has no surd in it.

  2. Choose what to multiply by

    22\dfrac{\sqrt{2}}{\sqrt{2}}

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

  3. Numerator

    1×2=21\times\sqrt{2} = \sqrt{2}

    The top becomes 2\sqrt{2}.

  4. Denominator

    2×2=2\sqrt{2}\times\sqrt{2} = 2

    The bottom becomes the whole number 2.

  5. Write the result

    22\dfrac{\sqrt{2}}{2}

    The denominator is now rational.

  6. Why not multiply by 2/2?

    12×22=222\dfrac{1}{\sqrt{2}}\times\dfrac{2}{2} = \dfrac{2}{2\sqrt{2}}

    That leaves a surd on the bottom, so it does not help.

  7. Confirm the value is unchanged

    22≈0.707\dfrac{\sqrt{2}}{2}\approx 0.707

    1/21/\sqrt{2} is also about 0.707.

  8. State the answer

    22\dfrac{\sqrt{2}}{2}

    So 1/2=2/21/\sqrt{2} = \sqrt{2}/2.

  9. Reason

    2×2=2\sqrt{2}\times\sqrt{2}=2

    Multiplying a surd by itself is what clears the root.

  10. Conclude

    22\dfrac{\sqrt{2}}{2}

    The correct rationalised form is 2/2\sqrt{2}/2.

  11. Underline the final answer

    =22= \dfrac{\sqrt{2}}{2}

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

  12. Check it answers the question

    22\dfrac{\sqrt{2}}{2}

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

  13. State the conclusion in one line

    22\dfrac{\sqrt{2}}{2}

    The working above leads directly to this result.

  14. Match to the options

    22\dfrac{\sqrt{2}}{2}

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

  15. Sense check

    22\dfrac{\sqrt{2}}{2}

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

Answer
22\dfrac{\sqrt{2}}{2}
Question 5
4 markschallenging
Which of the following is 72\sqrt{72} written in fully simplified surd form?
Show worked solution

Worked solution

  1. Find the largest square factor of 72

    72=36×272 = 36 \times 2

    36 is the biggest perfect square dividing 72.

  2. Why not 4 or 9?

    72=4×18=9×872 = 4\times18 = 9\times8

    These leave 18 or 8, which still contain square factors, so they are not fully simplified.

  3. Split using the largest square

    72=36×2\sqrt{72} = \sqrt{36}\times\sqrt{2}

    Separate the square part.

  4. Root the square part

    36=6\sqrt{36} = 6

    So 72=62\sqrt{72} = 6\sqrt{2}.

  5. Check the remainder

    2 is square-free2 \text{ is square-free}

    2\sqrt{2} cannot be simplified further.

  6. Reject 2182\sqrt{18}

    218=2×32=622\sqrt{18} = 2\times3\sqrt{2} = 6\sqrt{2}

    It equals the same value but is not fully simplified.

  7. Reject 383\sqrt{8}

    38=3×22=623\sqrt{8} = 3\times2\sqrt{2} = 6\sqrt{2}

    Again equal in value but not simplified.

  8. State the fully simplified form

    626\sqrt{2}

    Only 626\sqrt{2} has a square-free number under the root.

  9. Decimal check

    ≈8.485\approx 8.485

    626\sqrt{2} and 72\sqrt{72} both equal about 8.485.

  10. Conclude

    72=62\sqrt{72} = 6\sqrt{2}

    The correct fully simplified form is 626\sqrt{2}.

  11. Underline the final answer

    =62= 6\sqrt{2}

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

  12. Check it answers the question

    626\sqrt{2}

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

  13. State the conclusion in one line

    626\sqrt{2}

    The working above leads directly to this result.

  14. Match to the options

    626\sqrt{2}

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

  15. Sense check

    626\sqrt{2}

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

Answer
626\sqrt{2}

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