Pearson Edexcel International GCSE Mathematics A 4MA1: Integers, powers and roots – Revision Notes

Condensed for the final weeks. For full explanations and longer worked examples, use the Integers, powers and roots study guide.

These notes cover sections 1.1 (Integers) and 1.4 (Powers and roots) of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), first assessed in June 2018 with papers in January and June. Section 1.1 and the Foundation part of 1.4 are needed on both tiers. Surds and fractional powers (Higher statements 1.4A–C) are Higher tier only and are marked (H) below. A calculator is allowed on every 4MA1 paper, but show the method on every question here.

Links: practice questions, topic-level number revision notes, Edexcel IGCSE Mathematics hub, printable 4MA1 checklist, free 10-minute diagnostics.

1.1 Integers: definitions

Term	Meaning	Example
Integer	Whole number: positive, negative or zero	−4, 0, 17
Odd / even	Even divides exactly by 2; odd does not	0 is even
Factor of n	Divides into n with no remainder	6 is a factor of 54
Multiple of n	In the n times table	54 is a multiple of 6
Prime	Exactly two factors: 1 and itself	2, 3, 5, 7, 11, 13, …
Prime factor	A factor that is prime	prime factors of 54: 2, 3
Common factor	Divides both numbers	3 is a common factor of 12 and 21
Common multiple	In both times tables	42 is a common multiple of 6 and 7

1 is not prime. 2 is the only even prime.

Directed numbers

add a negative       ->  subtract      6 + (-10) = -4
subtract a negative  ->  add          -2 - (-7)  = 5
same signs (x or /)      ->  positive
different signs (x or /) ->  negative

In context: a rise is +, a fall is −. The difference between two temperatures is higher minus lower. From −12 °C to 3 °C is a rise of 3 − (−12) = 15 degrees.

Ordering: on a number line, further left is smaller. −20 < −3 even though 20 > 3.

Order of operations

1. Brackets
2. Powers (and roots)
3. Multiply and divide, LEFT to RIGHT
4. Add and subtract, LEFT to RIGHT

Worked reminder. 7 + 24 ÷ (5 − 9) = 7 + 24 ÷ (−4) = 7 + (−6) = 1.

Watch (−a)² against −a². (−4)² = 16 but −4² = −(4²) = −16. On a calculator, type the bracket.

1.4 Squares, cubes and roots

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Cubes to know: 1, 8, 27, 64, 125, and 1000.

• √ means the positive square root.
• Cube roots keep the sign: ∛(−8) = −2.
• 1 and 64 are both square and cube numbers.

Index laws (both tiers)

Law	Rule	Quick example
Multiply, same base	aᵐ × aⁿ = aᵐ⁺ⁿ	2⁶ × 2⁻⁴ = 2²
Divide, same base	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	10² ÷ 10⁵ = 10⁻³
Zero power	a⁰ = 1	13⁰ = 1
Negative power	a⁻ⁿ = 1/aⁿ	6⁻² = 1/36

Only combine powers of the same base. With a negative index, watch the signs: 5³ ÷ 5⁻² = 5³⁻⁽⁻²⁾ = 5⁵.

Prime factors, HCF and LCM

Method in steps: product of prime factors

1. Divide by the smallest prime that goes in (2, then 3, then 5, …).
2. Repeat on the answer until you reach 1.
3. Write the primes in index form, smallest first.
4. Check by multiplying back.

600 / 2 = 300, / 2 = 150, / 2 = 75, / 3 = 25, / 5 = 5, / 5 = 1
600 = 2³ × 3 × 5²

Method in steps: HCF and LCM

1. Write both numbers as products of prime powers.
2. HCF: primes common to both, lower power of each.
3. LCM: every prime in either, higher power of each.

Worked reminder. 90 = 2 × 3² × 5 and 126 = 2 × 3² × 7.

HCF = 2 × 3² = 18
LCM = 2 × 3² × 5 × 7 = 630
check: 18 × 630 = 11 340 = 90 × 126

A Venn diagram of the prime factors works too: the overlap multiplies to the HCF, and everything in the diagram multiplies to the LCM.

Which one does the question want?

Wording in the question	Use
“greatest number of identical groups”, “largest square tile”, “cut into equal lengths, none wasted”	HCF
“next time together”, “smallest number of packs so the totals match”, “first number in both lists”	LCM

After finding the HCF or LCM, reread the question. It often wants something you get from it: a clock time, a number of packs, or how many items go in each group.

Roots from prime factors. If every power in the prime factorisation is even, the number is a square. Halve each power to get the square root. If every power is a multiple of 3, the number is a cube. Divide each power by 3 to get the cube root.

3375 = 3³ × 5³   ->   ∛3375 = 3 × 5 = 15

Surds (H)

A surd is an irrational root, such as √3 or √10. √36 is not a surd.

√a × √b = √(ab)        √a × √a = a        √a / √b = √(a/b)
√(m²k) = m√k           (b + √a)(b - √a) = b² - a

• Simplify: take out the largest square factor. √98 = √(49 × 2) = 7√2.
• Add or subtract: only like surds. 4√3 − √3 = 3√3.
• Expand: every term times every term; √a × √a = a.
• Rationalise √a: multiply top and bottom by √a. 10/√5 = 10√5/5 = 2√5.
• Rationalise b ± √a: multiply top and bottom by the conjugate, b ∓ √a.

Fractional and negative powers (H)

a^(1/n)  = nth root of a
a^(m/n)  = (nth root of a)^m        root first
a^(-m/n) = 1 / a^(m/n)              flip
(a^m)^n  = a^(mn)                   both tiers (spec 2.1)

Worked reminder. 81^(3/4): fourth root of 81 is 3, and 3³ = 27.

For a fraction raised to a negative power, flip the fraction first, then apply the positive power.

Must-know distinctions

Don’t confuse	Difference
Factor and multiple	Factors of 12 are 12 or less (1, 2, 3, 4, 6, 12); multiples are 12 or more (12, 24, 36, …)
HCF and LCM	HCF ≤ the smaller number; LCM ≥ the larger number
a⁻ⁿ and −aⁿ	3⁻² = 1/9; −3² = −9
a^(1/2) and a/2	16^(1/2) = 4; 16/2 = 8
Prime factors and factors	Prime factors of 30 are 2, 3, 5; its factors are 1, 2, 3, 5, 6, 10, 15, 30
Surd and root	√5 is a surd; √25 = 5 is a root but not a surd

Quick self-test

1. Work out −14 − (−6) + 3.
2. Work out 40 − 3 × (2 − 6)².
3. Write down the two prime numbers between 50 and 60.
4. Write 1764 as a product of powers of its prime factors. Use your answer to find √1764.
5. Find the HCF and the LCM of 48 and 150.
6. Simplify 3⁴ × 3⁻⁶ ÷ 3⁻⁵ and evaluate it.
7. Work out ∛(−216).
8. (H) Simplify √12 + √75.
9. (H) Rationalise 3/(√7 − 2).
10. (H) Evaluate 64^(−2/3).
11. (H) Expand and simplify (3 − √2)².

Answers

1. −14 + 6 + 3 = −5.
2. (−4)² = 16; 3 × 16 = 48; 40 − 48 = −8.
3. 53 and 59.
4. 1764 = 2² × 3² × 7², so √1764 = 2 × 3 × 7 = 42 (halve each power).
5. 48 = 2⁴ × 3, 150 = 2 × 3 × 5². HCF = 2 × 3 = 6. LCM = 2⁴ × 3 × 5² = 1200.
6. Power: 4 + (−6) − (−5) = 3, so 3³ = 27.
7. (−6)³ = −216, so −6.
8. 2√3 + 5√3 = 7√3.
9. Multiply by (√7 + 2)/(√7 + 2): denominator 7 − 4 = 3, so 3(√7 + 2)/3 = √7 + 2.
10. 64^(1/3) = 4, 4² = 16, flip: 1/16.
11. 9 − 6√2 + 2 = 11 − 6√2.

Where marks are usually lost

• Including 1 in a list of primes, or leaving out 2.
• Giving 2 × 2 × 2 × 3 × 5 × 5 when the question asks for a product of powers of prime factors.
• Using the higher powers for the HCF, or only the common primes for the LCM.
• Answering an LCM context question with the LCM itself when the question asks for a clock time or a number of packs.
• Working left to right across + and × together: 7 + 24 ÷ (−4) is 1, not −7.75.
• Writing 6⁻² = −36 or −12.
• Adding unlike surds: √12 + √75 is not √87.
• (H) Expanding (3 − √2)² as 9 + 2, missing the −6√2 term.
• (H) Rationalising with the same sign instead of the conjugate.
• (H) Applying the power before the root, producing huge numbers and arithmetic slips.

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).