AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 19

5 marks · Medium difficulty · Short Answer

Work out the value of (9/16)^(-3/2) and find the value of n such that sqrt(125) = 5^n.

Practise this question

Question

Question 19 contains two parts: Part (a) asks to work out the value of (9/16) raised to the power of -3/2, worth 3 marks. Part (b) states that the square root of 125 equals 5 to the power of n, and asks to work out the value of n, worth 2 marks.
Question text

−

19 (a) Work out the value of

[3 marks]

Answer

19 (b) 125 = 5n

Work out the value of n.

[2 marks]

Mark scheme

Show the mark scheme Mark scheme for question 19. For 19(a): answer 64/27 or 2 and 10/27 (B3). Partial credit: B2 for showing understanding of at least two of negative index (reciprocal), index 1/2 (square root), or index 3 (cube); B1 for showing understanding of one. For 19(b): answer 1.5, 3/2, or 1 and 1/2 (B2). Partial credit B1 for 125^(1/2), (5^3)^(1/2), (5^(1/2))^3, 5^(3/2), 125 = 5^(2n), 5^3 = 5^(2n), or 2n = 3.

Q Answer Mark Comments

64 10 B2

or 2

27 27 shows understanding of at least two of

• a negative index means reciprocal

B3 • an index of means square root

• an index of 3 means cube

B1

shows understanding of one of the above

Additional Guidance

Condone missing brackets

Allow equivalent decimals, fractions or mixed numbers throughout

Ignore any attempt to simplify, change to a mixed number or convert to a

decimal after a correct answer is seen

64 20 B3

eg = 2

27 27

Understanding can be shown in expressions which are not equivalent to the

19(a)

original

27 B2

eg implies understanding of square root and cubing, but not reciprocal

1 −1

9 4096 2 27

Example B2 expressions: or or B2

1 729 64

−3 −

16 3 729 2

Example B1 expressions: or or B1

94 4096

To show understanding, correct method or value for calculation of a cube is

required, but calculation of a square root is not required

eg1 only shows understanding of the square root B1

99 9

eg2 × × shows understanding of the square root B2

16 16 16

and cubing

31 1 1 1 3 3

1.5 or or 1 3 2 2

22 B1 125 2 or (5 ) or 5 or 52

19(b) B2

or

125 = 52n or 53 = 52n or 2n = 3

How to answer it

Laws of Indices: Fractional & Negative Powers

📋 What this question tests

This question assesses your mastery of Higher Tier index laws: evaluating expressions with negative fractional powers, interpreting roots as fractional powers, writing numbers as powers of a common base, and solving simple exponential equations.

Question 19 (a)

Work out the value of (9/16)−3/2  [3 marks]

📐 Step-by-Step Calculation

Break the index −3/2 down into three distinct operations:

  1. Negative power: Take the reciprocal (flip the fraction):
    (9/16)−3/2 = (16/9)3/2
  2. Denominator of 2: Take the square root:
    √(16/9) = 4/3
  3. Numerator of 3: Cube the result:
    (4/3)³ = (4 × 4 × 4) / (3 × 3 × 3) = 64/27

✅ Correct Answer

64/27  or  2 10/27

Mark Breakdown:
• B3: Fully correct value ( 64/27 or 2 10/27 ).
• B2: Shows understanding of any two rules (e.g. 27/64 shows root and cube, but not reciprocal; or (27/64)⁻¹ ).
• B1: Shows understanding of any one rule (e.g. 16/9 , (3/4)⁻³ , or (√(9/16))³ ).

💡 Key Knowledge

  • Negative index: (a/b)⁻ⁿ = (b/a)ⁿ
  • Fractional index: a1/n = ⁿ√a
  • Combined power: am/n = (ⁿ√a)ᵐ
  • Always root before you power! It keeps the numbers small and easy to calculate mentally.

❌ Common Errors

  • Multiplying instead of powering: Writing 9/16 × (−3/2) . Powers are NOT scale factors.
  • Forgetting to invert: Getting 27/64 by finding the power of 3/2 but forgetting the negative sign.
  • Arithmetic slips on cubing: Calculating 4³ as 12 instead of 64, or 3³ as 9 instead of 27.

Question 19 (b)

√125 = 5n  —  Work out the value of n  [2 marks]

📐 Step-by-Step Calculation

Express both sides with a common base of 5:

  1. Method 1 (Base form inside root):
    Write 125 as a power of 5: 125 = 5³
    Apply the square root rule ( √x = x1/2 ):
    √125 = √(5³) = (5³)1/2
    Multiply powers: 3 × 1/2 = 3/2
    Therefore, 53/2 = 5n ⇒ n = 3/2 (or 1.5)
  2. Method 2 (Square both sides):
    125 = (5n)²
    5³ = 52n
    Equate indices: 2n = 3 ⇒ n = 3/2

✅ Correct Answer

n = 3/2  or  1.5  or  1 1/2

Mark Breakdown:
• B2: Correct value of n ( 1.5 or equivalent fraction).
• B1: Partial credit for writing any of:
  – 1251/2
  – (5³)1/2 or (51/2)³ or 53/2
  – 125 = 52n or 5³ = 52n or 2n = 3

🧠 Exam Technique

  • Look for the prime base: The RHS has a base of 5. Immediately rewrite 125 in prime base form: 125 = 5³ .
  • Equating powers: Once both sides have base 5, you can simply set the powers equal: 5ᵃ = 5ᵇ ⇒ a = b .
  • Check your work: 51.5 = 5¹ × 50.5 = 5√5. Since √125 = √(25 × 5) = 5√5, this verifies n = 1.5 is correct.

❌ Common Errors

  • Thinking √ means ÷ 2: Writing √125 as 62.5 . Square root is the inverse of squaring, not halving.
  • Halving the base: Confusing (5³)1/2 with 2.5³ .
  • Leaving the answer as an expression: Giving 53/2 on the answer line rather than stating the value of n itself ( 3/2 ).

Topics

Number · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.