Pearson Edexcel International GCSE Mathematics A 4MA1: Integers, powers and roots – 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 cover sections 1.1 (Integers) and 1.4 (Powers and roots) of topic 1, Numbers and the number system, in the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), first assessed in June 2018 with papers in January and June. Questions 1 to 8 and 12 use content that both tiers need. Questions 9, 10 and 11 are marked (Higher): they use the Higher tier only statements on surds and fractional and negative powers. A calculator may be used on every 4MA1 paper, but these questions test exact methods, so show every step.

Related: study guide, revision notes, mixed topic 1 practice, course hub, printable checklist.

Questions

1. On Thursday the lowest temperature in a town was −11 °C and the highest was 6 °C.

(a) Work out the difference between the highest and lowest temperatures. [1] (b) On Friday the lowest temperature was 4 degrees lower than on Thursday. Write down Friday’s lowest temperature. [1]

2. This question is about directed numbers.

(a) Write these numbers in order of size, smallest first: 7, −12, 0, −3, 5, −20 [1] (b) Work out (−8) × (−5) − 6 × (−3). [2]

3. This question is about the order of operations.

(a) Work out 52 − 3 × (9 − 14)² ÷ 5. Show your working. [3] (b) Copy 6 + 18 ÷ 3 − 1 = 15 and insert one pair of brackets to make it correct. [1]

4. Here is a list of numbers: 9, 15, 23, 27, 32, 39, 41, 64. From the list, write down

(a) the prime numbers [1] (b) the multiples of 3 [1] (c) the number that is both a square number and a cube number [1] (d) the number that is a factor of 96. [1]

5. This question is about prime factors.

(a) Write 1188 as a product of powers of its prime factors. [3] (b) 990 = 2 × 3² × 5 × 11. Find the highest common factor (HCF) and the lowest common multiple (LCM) of 1188 and 990. [3]

6. Two warning lights flash together at exactly 21:00. One light flashes every 40 seconds. The other flashes every 75 seconds. Work out the time when they next flash together. Show your working. [4]

7. This question is about index laws.

(a) Write 8⁵ × 8⁻⁷ as a single power of 8. [1] (b) Work out 4⁰ + 4⁻¹, giving your answer as a fraction. [2] (c) 9ⁿ × 9³ ÷ 9⁸ = 9⁻². Find the value of n. [2]

8. This question is about squares, cubes and roots.

(a) Work out √(13² − 5²). [2] (b) Find the square number between 300 and 400 that is a multiple of 3. [2] (c) Work out ∛(−512). [1]

9. (Higher) Give each answer in its simplest surd form.

(a) Simplify √200 − √18. [2] (b) Expand and simplify (4 + √5)(2 − √5). [2] (c) Rationalise the denominator of 14/√7. [2]

10. (Higher) Show your working. A calculator answer alone scores nothing.

(a) Evaluate 125^(2/3). [2] (b) Evaluate (16/81)^(−3/4). [3] (c) Work out 2⁻¹ × 8^(4/3). [2]

11. (Higher)

(a) Show that (√6 + √2)/(√6 − √2) = 2 + √3. [4] (b) A rectangle has sides of length (3 + √2) cm and (3 − √2) cm. (i) Work out the area of the rectangle. [2] (ii) Work out the exact length of a diagonal of the rectangle. [3]

12. A florist has 156 roses and 117 tulips. She makes identical bunches. Each bunch has the same number of roses and the same number of tulips, and every flower is used.

(a) Write 156 and 117 as products of powers of their prime factors. [2] (b) Find the greatest number of bunches she can make, and the number of roses and tulips in each bunch. [3] (c) Roses come in packs of 24 and tulips in packs of 18. Next week she wants to buy exactly the same number of roses as tulips, using whole packs. Find the smallest number of packs of each she can buy. [3]

Answers

1. (a) 6 − (−11) = 17 degrees [1] (b) −11 − 4 = −15 °C [1] Examiner insight: A bare answer of 5 for part (a) comes from 11 − 6 and scores nothing; the sign of −11 must be kept.

2. (a) −20, −12, −3, 0, 5, 7 [1] (b) (−8) × (−5) = 40 and 6 × (−3) = −18 [1]; 40 − (−18) = 58 [1] Examiner insight: The method mark needs both products correct with their signs; 40 − 18 = 22 loses the accuracy mark.

3. (a) (9 − 14)² = (−5)² = 25 [1]; 3 × 25 ÷ 5 = 15 [1]; 52 − 15 = 37 [1] (b) 6 + 18 ÷ (3 − 1) = 15 [1] Examiner insight: Working 52 − 3 first (giving 49 × 25 ÷ 5 = 245) breaks the hierarchy and gets no credit after the first step.

4. (a) 23, 41 [1] (b) 9, 15, 27, 39 [1] (c) 64 [1] (d) 32 [1] Examiner insight: A list mark is usually lost for any extra wrong entry, so including 27 or 39 as primes costs the mark even if 23 and 41 are there.

5. (a) Repeated division: 1188 ÷ 2 = 594, ÷ 2 = 297, ÷ 3 = 99, ÷ 3 = 33, ÷ 3 = 11 [1]; all five primes found: 2, 2, 3, 3, 3, 11 [1]; 2² × 3³ × 11 [1] (b) HCF = 2 × 3² × 11 [1] = 198 [1]; LCM = 2² × 3³ × 5 × 11 = 5940 [1] Examiner insight: For (a), the final mark needs index form; 2 × 2 × 3 × 3 × 3 × 11 gets the method marks only.

6. 40 = 2³ × 5 and 75 = 3 × 5² [1]; LCM = 2³ × 3 × 5² [1] = 600 seconds [1]; 600 s = 10 minutes, so 21:10 [1] Examiner insight: Stopping at 600 loses the final mark: the question asks for a time, and 21:06 (from reading 600 seconds as 6 minutes) is also wrong.

7. (a) 8⁻² [1] (b) 4⁰ = 1 and 4⁻¹ = 1/4 [1]; total 5/4 (or 1¼) [1] (c) n + 3 − 8 = −2 [1]; n = 3 [1] Examiner insight: In (b), writing 4⁰ = 0 is a common slip and loses both marks, since the method mark needs both terms right.

8. (a) 169 − 25 = 144 [1]; √144 = 12 [1] (b) Square numbers between 300 and 400 are 324 and 361 [1]; 324 = 3 × 108, so 324 [1] (c) (−8)³ = −512, so −8 [1] Examiner insight: In (a), √169 − √25 = 13 − 5 = 8 is a wrong method and scores zero, even with correct arithmetic.

9. (a) √200 = 10√2 and √18 = 3√2 [1]; 7√2 [1] (b) 8 − 4√5 + 2√5 − 5 [1]; 3 − 2√5 [1] (c) 14√7/7 [1] = 2√7 [1] Examiner insight: Decimal answers (such as 9.899 for (a)) get no credit where the question asks for simplest surd form.

10. (a) ∛125 = 5 [1]; 5² = 25 [1] (b) Flip: (81/16)^(3/4) [1]; fourth root: 3/2 [1]; cube: 27/8 [1] (c) 8^(4/3) = 2⁴ = 16 [1]; 16 × 1/2 = 8 [1] Examiner insight: In (b) the flip is the first method mark; an answer of 8/27 shows the reciprocal was forgotten and scores 1 at most.

11. (a) Multiply top and bottom by (√6 + √2) [1]; denominator 6 − 2 = 4 [1]; numerator (√6 + √2)² = 6 + 2√12 + 2 = 8 + 2√12 [1]; √12 = 2√3, so (8 + 4√3)/4 = 2 + √3 [1] (b)(i) (3 + √2)(3 − √2) = 9 − 2 [1] = 7 cm² [1] (ii) d² = (3 + √2)² + (3 − √2)² [1]; = (11 + 6√2) + (11 − 6√2) = 22 [1]; d = √22 cm [1] Examiner insight: On a “show that”, every line must be shown; jumping from (8 + 2√12)/4 to 2 + √3 without simplifying √12 loses the final mark.

12. (a) 156 = 2² × 3 × 13 [1]; 117 = 3² × 13 [1] (b) HCF = 3 × 13 [1] = 39 bunches [1]; each bunch has 4 roses and 3 tulips [1] (c) LCM of 24 and 18: 24 = 2³ × 3, 18 = 2 × 3² [1]; LCM = 2³ × 3² = 72 [1]; 72 ÷ 24 = 3 and 72 ÷ 18 = 4, so 3 packs of roses and 4 packs of tulips [1] Examiner insight: Part (c) needs the numbers of packs; answering 72 alone earns the method marks but not the final mark.

Where marks are usually lost

• Dropping the negative sign when finding a difference between temperatures.
• Working left to right across + and × instead of following the order of operations.
• Writing products of primes without index form when “product of powers” is asked for.
• Mixing up HCF (lower powers, common primes) with LCM (higher powers, all primes).
• Stopping at the LCM or HCF when the context asks for a time, number of packs or contents of a bunch.
• Treating a⁰ as 0, or a⁻ⁿ as a negative number.
• (Higher) Adding or subtracting surds before simplifying them to the same surd.
• (Higher) Forgetting the middle term when squaring a bracket containing a surd.
• (Higher) Forgetting to flip the fraction for a negative fractional power.

Next steps

• Revision notes for this unit
• Study guide for this unit
• Edexcel IGCSE Mathematics hub
• Printable 4MA1 checklist
• All free 10-minute diagnostics
• Book a free trial class

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification Issue 2, November 2017, Pearson Education Limited. Topic 1, Numbers and the number system: 1.1 Integers and 1.4 Powers and roots (Foundation Tier content, with Higher Tier statements 1.4A–C on surds and fractional and negative powers).