Pearson Edexcel International GCSE Mathematics A 4MA1: Integers, powers and roots – Study Guide

This study guide teaches 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 available in January and June. Section 1.1 is Foundation tier content that the Higher tier assumes (“See Foundation Tier”). Section 1.4 has Foundation content that both tiers need, plus three extra statements (surds, rationalising, and fractional and negative powers) that are Higher tier only.

Each tier sits two papers (1F and 2F, or 1H and 2H). Each paper lasts 2 hours, carries 100 marks and is worth 50% of the qualification. A calculator may be used on every paper. Even so, questions on primes, index laws and surds expect exact working, and a calculator answer with no method often scores nothing on them.

See also the revision notes, the practice questions, the topic-level Numbers and the number system study guide, the Edexcel IGCSE Mathematics hub, the printable 4MA1 checklist and the free 10-minute diagnostics.

What this unit covers

Section	What you must be able to do	Tier
1.1 A–B	Understand and use integers (positive, negative and zero); understand place value	Both tiers
1.1 C–D	Use directed numbers in practical situations (for example temperatures); order integers	Both tiers
1.1 E–F	Use the four rules; use brackets and the hierarchy of operations	Both tiers
1.1 G–H	Use the terms odd, even, prime numbers, factors and multiples; identify prime factors, common factors and common multiples	Both tiers
1.4 A–B	Identify square and cube numbers; calculate squares, square roots, cubes and cube roots	Both tiers
1.4 C	Use index notation and index laws for multiplication and division of positive and negative integer powers, including zero	Both tiers
1.4 D–E	Express integers as a product of powers of prime factors; find HCF and LCM	Both tiers
1.4 A (H)	Understand the meaning of surds	Higher tier only
1.4 B (H)	Manipulate surds, including rationalising a denominator	Higher tier only
1.4 C (H)	Use index laws to simplify and evaluate numerical expressions with integer, fractional and negative powers	Higher tier only

1.1 Integers

Place value and ordering

An integer is a whole number: positive, negative or zero. Place value tells you what each digit is worth. In 3 470 215, the 4 is worth 400 000 and the 7 is worth 70 000.

To order integers, picture a number line. The further left, the smaller. A negative number is always smaller than zero, and −15 is smaller than −2 because it lies further left.

Worked example. Write −8, 3, −15, 0, −2, 11 in order, smallest first.

negatives first, largest "size" first:  -15, -8, -2
then zero and positives:                 0, 3, 11
answer: -15, -8, -2, 0, 3, 11

Directed numbers in context

Temperatures, bank balances and heights above and below sea level all use negative numbers. A rise means add; a fall means subtract. The difference between two values is the larger minus the smaller.

Worked example. At 05:00 the temperature in a mountain village is −9 °C. By 14:00 it has risen by 13 degrees. By 22:00 it has fallen by 6 degrees from the 14:00 value. Find the temperature at 14:00 and at 22:00, and the difference between the 05:00 and 14:00 temperatures.

14:00:   -9 + 13 = 4 °C
22:00:    4 - 6  = -2 °C
difference:  4 - (-9) = 4 + 9 = 13 degrees

The four rules with negative numbers

• Subtracting a negative is the same as adding: −5 − (−9) = −5 + 9 = 4.
• Adding a negative is the same as subtracting: −3 + (−8) = −11.
• Multiplying or dividing: same signs give a positive, different signs give a negative. (−6) × (−4) = 24 and (−36) ÷ 9 = −4.

Brackets and the hierarchy of operations

Work in this order: brackets, then powers, then multiplication and division (left to right), then addition and subtraction (left to right).

Worked example. Work out 18 − 4 × (7 − 10)² ÷ 6.

brackets:          7 - 10 = -3
powers:            (-3)^2 = 9
multiply/divide:   4 x 9 = 36,  36 / 6 = 6
subtract:          18 - 6 = 12

Multiplication and division have equal priority, so work left to right: 30 ÷ (2 + 3) × 2 = 6 × 2 = 12, not 30 ÷ 10 = 3.

Odd, even, primes, factors and multiples

• An even number divides exactly by 2; an odd number does not.
• A factor of n divides into n exactly. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
• A multiple of n is in the n times table: the multiples of 7 are 7, 14, 21, …
• A prime number has exactly two factors, 1 and itself. The primes between 30 and 50 are 31, 37, 41, 43, 47.
• A prime factor is a factor that is prime. The prime factors of 42 are 2, 3 and 7.
• A common factor of two numbers divides both; a common multiple is in both times tables.

1.4 Powers and roots

Square and cube numbers

Learn these so you recognise them on sight:

squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
cubes:   1, 8, 27, 64, 125, and 1000 (= 10^3)

Squaring and square-rooting undo each other, as do cubing and cube-rooting. So 13² = 169 and √196 = 14; 6³ = 216 and ∛343 = 7. The √ sign means the positive root. A cube root of a negative number is negative: ∛(−125) = −5, because (−5)³ = −125. A negative number has no real square root.

Index notation and index laws

Index notation is shorthand for repeated multiplication: 5⁴ = 5 × 5 × 5 × 5. Both tiers need these laws for integer powers:

Law	Rule	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	5⁷ × 5⁻³ = 5⁴ = 625
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	7³ ÷ 7⁸ = 7⁻⁵
Zero power	a⁰ = 1	9⁰ = 1
Negative power	a⁻ⁿ = 1/aⁿ	4⁻² = 1/4² = 1/16

The laws only work when the base is the same. 2³ × 3⁴ cannot be written as a single power.

Worked example. Simplify (3⁵ × 3⁻²) ÷ 3⁴, giving the answer as a fraction.

numerator:   3^(5 + (-2)) = 3^3
divide:      3^(3 - 4) = 3^-1
value:       3^-1 = 1/3

Product of prime factors

Every integer greater than 1 can be written as a product of primes in exactly one way, apart from the order. Use a factor tree or repeated division by primes, starting with the smallest.

Worked example. Write 1260 as a product of powers of its prime factors.

1260 / 2 = 630
 630 / 2 = 315
 315 / 3 = 105
 105 / 3 = 35
  35 / 5 = 7
   7 / 7 = 1
1260 = 2 x 2 x 3 x 3 x 5 x 7 = 2² × 3² × 5 × 7

“Product of powers” means you must use index form for repeated primes. Check by multiplying back: 4 × 9 × 5 × 7 = 1260.

Highest common factor and lowest common multiple

Write each number as a product of primes, then:

• HCF: multiply the primes that appear in both, each to the lower power.
• LCM: multiply every prime that appears in either, each to the higher power.

Worked example. Find the HCF and LCM of 84 and 120.

84  = 2² × 3 × 7
120 = 2³ × 3 × 5
HCF = 2² × 3 = 12
LCM = 2³ × 3 × 5 × 7 = 840

A check: HCF × LCM = 12 × 840 = 10 080 = 84 × 120. This works for any two numbers.

Recognising the context. “Largest”, “greatest”, or “split into equal groups with none left over” points to the HCF. “Next time they happen together”, “smallest number both go into” points to the LCM. Two buses leave a station together at 08:00; one runs every 12 minutes and the other every 18 minutes. LCM(12, 18) = 36, so they next leave together at 08:36.

1.4 Surds (Higher tier only)

What a surd is

A surd is a root that cannot be written as an exact fraction, such as √2, √5 or ∛4. Surds are irrational: their decimals never end or repeat. √49 = 7 is not a surd. Leaving an answer as a surd keeps it exact, and exam questions often ask for exactly that.

Rules for manipulating surds

√a × √b = √(ab)            √a ÷ √b = √(a/b)
√a × √a = a                 k√a + m√a = (k + m)√a

There is no rule for √a + √b: it is not √(a + b).

Simplifying. Find the largest square factor. √72 = √(36 × 2) = 6√2.

Adding. Simplify first so the surds match: √45 + √20 = 3√5 + 2√5 = 5√5.

Expanding brackets. Multiply every term by every term, then use √3 × √3 = 3.

(2 + √3)(4 - √3) = 8 - 2√3 + 4√3 - 3
                 = 5 + 2√3
(1 + √7)²        = 1 + 2√7 + 7 = 8 + 2√7

Note that (1 + √7)² is not 1 + 7. You must include the middle term.

Rationalising a denominator

To rationalise means to rewrite a fraction so that the denominator has no surd.

• Denominator √a: multiply top and bottom by √a.
• Denominator b + √a: multiply top and bottom by b − √a (and the reverse for b − √a). This uses (b + √a)(b − √a) = b² − a, which has no surd.

Worked example. Rationalise (a) 6/√3 and (b) 4/(3 + √5).

(a) 6/√3 = (6 × √3)/(√3 × √3) = 6√3/3 = 2√3

(b) 4/(3 + √5) × (3 - √5)/(3 - √5)
    denominator: 3² - 5 = 4
    numerator:   4(3 - √5)
    answer:      4(3 - √5)/4 = 3 - √5

1.4 Fractional and negative powers (Higher tier only)

At Higher tier you also evaluate expressions with fractional powers. Along with the laws above, use the power-of-a-power law (aᵐ)ⁿ = aᵐⁿ, which the specification lists under algebra (section 2.1).

a^(1/n) = nth root of a              e.g. 32^(1/5) = 2
a^(m/n) = (nth root of a)^m          root first keeps numbers small
a^(-m/n) = 1/a^(m/n)                 negative means reciprocal

Worked example. Evaluate (a) 27^(2/3), (b) 16^(−3/4), (c) (8/27)^(−2/3).

(a) 27^(1/3) = 3,  then 3² = 9

(b) 16^(1/4) = 2,  2³ = 8,  negative power: 1/8

(c) flip for the negative power: (27/8)^(2/3)
    cube root: 3/2,  square: 9/4

Doing the root first matters. 32^(3/5): the fifth root of 32 is 2, and 2³ = 8. Cubing first gives 32 768, whose fifth root is much harder to find without a calculator.

Common errors

• Calling 1 a prime number. It has only one factor.
• Writing a product of primes without powers (2 × 2 × 3 × 3 × 5 × 7) when the question says “product of powers of prime factors”.
• Swapping HCF and LCM: taking the higher powers for the HCF.
• Working 30 ÷ 2 × 5 as 30 ÷ 10. Division and multiplication go left to right.
• Squaring a negative inside a bracket wrongly: (−3)² = 9, but −3² = −9.
• Reading 4⁻² as −16 or −8. A negative power means reciprocal.
• Adding powers with different bases: 2³ × 3⁴ ≠ 6⁷.
• Writing √45 + √20 = √65.
• Rationalising 4/(3 + √5) by multiplying by (3 + √5) instead of the conjugate.
• Forgetting to flip the fraction when the power is negative and fractional.

Where to go next

Recap with the revision notes, then try the practice questions. For mixed questions across the whole of topic 1, use the Numbers and the number system practice questions and the topic-level revision notes.

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).