OxfordAQA International GCSE Mathematics: Index laws, surds, standard form and sets (9260) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions are aligned to the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards), statements N6 to N9 in section 3.1.1. Core content is also assessable on the Extension tier. Questions marked (Extension only) test fractional powers or surds. Learn the methods in the study guide and the revision notes first. See also the course hub, the printable checklist and the number practice set.

Questions

1. Simplify each expression, giving positive indices in your answers.

(a) 7c³ × 3c⁻⁵ [2] (b) 20d⁶ ÷ 8d² [2]

2. (a) Work out the value of 2⁻⁴. [1] (b) Work out the value of (3/2)⁻³ as a fraction. [2] (c) Solve 3^(x+2) = 1/81. [2]

3. (Extension only) Work out the value of each, giving fractions where needed.

(a) 125^(2/3) [2] (b) 16^(−3/4) [2] (c) Simplify (49a⁶)^(3/2). [2]

4. Work out each, with the answer as a simplified fraction. Do not use decimals.

(a) 3⅛ − 1⅚ [3] (b) (3/4 + 5/9) ÷ 1⅙ [3]

5. (Extension only)

(a) Show √288 can be written as a√2 and state the whole number a. [1] (b) Simplify 5√18 − 2√50. [3] (c) Expand and simplify (5 − √6)². [2]

6. (Extension only)

(a) Rationalise the denominator of 14/√7. [2] (b) Write 12/(5 − √13) in the form p + √q, where p and q are integers. [3] (c) Show that (2 + √5)/(√5 − 1) = (7 + 3√5)/4. [4]

7. (Extension only) A right-angled triangle has perpendicular sides of length 2√6 cm and 3√2 cm.

(a) Show that its area is 6√3 cm². [3] (b) Find the exact length of the hypotenuse. [2]

8. (a) Write 0.000 0407 in standard form. [1] (b) Write 3.6 × 10⁷ as an ordinary number. [1] (c) Work out (2.5 × 10⁻³) × (6.4 × 10⁶), giving your answer in standard form. [2] (d) Work out (9.6 × 10⁵) + (4.4 × 10⁶), giving your answer in standard form. [2]

9. A server holds 2.4 × 10¹⁴ bytes of data. Each stored file is 6 × 10⁵ bytes.

(a) How many files can the server hold? Give your answer in standard form. [2] (b) A cell has a width of 1.2 × 10⁻⁷ m. A line of 5 × 10⁴ of these cells is laid end to end. Find the length of the line in metres, in standard form. [2] (c) A calculator shows 3.15E-4 for another result. Write this in standard form and as an ordinary decimal. [2]

10. ξ = {integers from 1 to 30}. A = {factors of 30}. B = {multiples of 5}.

(a) List the elements of A ∩ B. [2] (b) Find n(A ∪ B). [2] (c) Find n(B′). [1] (d) Find n((A ∪ B)′). [1]

11. Of 70 people at a gym, 30 use the treadmill (T), 28 use the rowing machine (R) and 34 use the weights (W). 10 use T and R, 12 use R and W, 11 use T and W, and 5 use all three.

(a) Draw a Venn diagram and write the number in each of the eight regions. [4] (b) How many people use exactly one of the three? [1] (c) How many people use the weights but not the rowing machine? [2]

12. In a class of 36 pupils, 20 study French (F), 15 study Spanish (S) and 5 study neither. x pupils study both.

(a) Form and solve an equation to find x. [3] (b) How many pupils study French but not Spanish? [1]

Answers

1. (a) Numbers: 7 × 3 = 21 [1]. Indices: 3 + (−5) = −2, so 21/c² [1] (b) 20 ÷ 8 = 5/2 [1]. 6 − 2 = 4, so 5d⁴/2 [1] [4] Examiner insight: Show the coefficient work and the index work as two separate lines, so a slip in one does not hide that the other was right.

2. (a) 1/16 [1] (b) Flip the fraction: (2/3)³ [1] = 8/27 [1] (c) 1/81 = 3⁻⁴ [1]. So x + 2 = −4 and x = −6 [1] [5] Examiner insight: In part (c), writing the right-hand side as a power of 3 is the key line to show. A trial-and-error answer with no working is a risk on a two-mark question.

3. (a) 125^(1/3) = 5 [1]. 5² = 25 [1] (b) 16^(1/4) = 2 [1]. Negative power: 2⁻³ = 1/8 [1] (c) 49^(3/2) = 7³ = 343 [1]. (a⁶)^(3/2) = a⁹, so 343a⁹ [1] [6] Examiner insight: Taking the root first keeps the numbers small. Remember to apply the power to the letter as well as the number.

4. (a) 25/8 − 11/6 [1] = 75/24 − 44/24 [1] = 31/24 (or 1⁷⁄₂₄) [1] (b) 3/4 + 5/9 = 27/36 + 20/36 = 47/36 [1]. 1⅙ = 7/6, so 47/36 × 6/7 [1] = 282/252 = 47/42 (or 1⁵⁄₄₂) [1] [6] Examiner insight: When the question says “fraction” and “no decimals”, a rounded decimal gives away the exact answer. Converting the mixed numbers first is the line that shows your method.

5. (a) √288 = √(144 × 2) = 12√2 [1] (b) 5√18 = 5 × 3√2 = 15√2 [1]. 2√50 = 2 × 5√2 = 10√2 [1]. Difference: 5√2 [1] (c) 25 − 10√6 + 6 [1] = 31 − 10√6 [1] [6] Examiner insight: A common slip is squaring only the two terms and writing 25 + 6. Write all four products before collecting terms.

6. (a) Multiplying both parts by √7 gives 14√7/7 [1] = 2√7 [1] (b) Use the conjugate: multiply numerator and denominator by (5 + √13) [1]. Denominator: 25 − 13 = 12 [1]. 12(5 + √13)/12 = 5 + √13 [1] (c) Multiply numerator and denominator by (√5 + 1) [1]. Numerator: 2√5 + 2 + 5 + √5 [1]. Denominator: 5 − 1 = 4 [1]. Numerator = 7 + 3√5, so the fraction is (7 + 3√5)/4 as required [1] [9] Examiner insight: On a “show that” question, every line must be visible, including the expanded numerator and the denominator calculation. Stating the result without these lines is unlikely to be enough.

7. (a) Area = ½ × 2√6 × 3√2 [1] = 3√12 [1] = 3 × 2√3 = 6√3 cm² [1] (b) h² = (2√6)² + (3√2)² = 24 + 18 [1]. h = √42 cm [1] [5] Examiner insight: Keep the answer exact. A decimal such as 6.48 does not show the surd the question asks for, and rounding early can change the final digits.

8. (a) 4.07 × 10⁻⁵ [1] (b) 36 000 000 [1] (c) 2.5 × 6.4 = 16 and 10⁻³ × 10⁶ = 10³ [1]. 16 × 10³ = 1.6 × 10⁴ [1] (d) 9.6 × 10⁵ = 0.96 × 10⁶ [1]. 0.96 × 10⁶ + 4.4 × 10⁶ = 5.36 × 10⁶ [1] [6] Examiner insight: A result like 16 × 10³ is not yet in standard form. Check 1 ≤ A < 10 before you write the final answer.

9. (a) (2.4 ÷ 6) × 10¹⁴⁻⁵ = 0.4 × 10⁹ [1] = 4 × 10⁸ files [1] (b) 1.2 × 5 = 6 and 10⁻⁷ × 10⁴ = 10⁻³ [1]. Length = 6 × 10⁻³ m [1] (c) 3.15 × 10⁻⁴ [1], which is 0.000 315 [1] [6] Examiner insight: Include the unit in context questions. Copying “3.15E-4” as your final answer shows the display and not the number.

10. (a) A = {1, 2, 3, 5, 6, 10, 15, 30} and B = {5, 10, 15, 20, 25, 30} [1]. A ∩ B = {5, 10, 15, 30} [1] (b) n(A) = 8, n(B) = 6, n(A ∩ B) = 4 [1]. 8 + 6 − 4 = 10 [1] (c) 30 − 6 = 24 [1] (d) 30 − 10 = 20 [1] [6] Examiner insight: Listing the elements of both sets first makes later parts quick and checkable. A′ and B′ ask for what is outside, so subtract from the total.

11. (a) All three: 5 [1]. Pairs only: T and R = 5, R and W = 7, T and W = 6 [1]. Single sets: T only = 14, R only = 11, W only = 16 [1]. Outside: 70 − 64 = 6 [1] (b) 14 + 11 + 16 = 41 [1] (c) W only plus T and W only: 16 + 6 [1] = 22 [1] [7] Examiner insight: Begin in the centre and work outwards. Check that the eight regions add to 70 before answering parts (b) and (c).

12. (a) French only = 20 − x, Spanish only = 15 − x [1]. (20 − x) + x + (15 − x) + 5 = 36 [1]. 40 − x = 36, so x = 4 [1] (b) 20 − 4 = 16 [1] [4] Examiner insight: Label the regions in terms of x before forming the equation. Writing “20 + 15 + 5 = 36” without subtracting the overlap is the usual route to a wrong answer.

Where marks are usually lost

• Reading a negative index as a negative value, for example 2⁻⁴ = −16.
• Multiplying indices instead of adding them when multiplying powers of the same base.
• Applying a fractional power to the number but not to the letter.
• Rounding in the middle of a fraction or surd calculation when the question asks for an exact answer.
• Not simplifying a surd fully, or not rationalising the denominator when told to.
• Dropping the middle terms when squaring a bracket with a surd.
• Leaving a standard-form answer as 16 × 10³, or adding standard-form numbers without matching the powers of 10.
• Copying a calculator display (3.15E-4) instead of writing the full standard form.
• Counting the overlap twice in a set problem, or filling the single-set regions first.

Next steps

• Revisit the study guide for any question you missed.
• Use the revision notes for a quick recap.
• Tick off progress on the printable checklist and see the course hub.
• Take the free 10-minute diagnostics to find the next topic to work on.
• If you would like help from a teacher, book a free trial class.

Official syllabus

Source: OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Section 3.1.1, statements N6 to N9.