Number, Indices & Standard Form

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Corbett MathsProduct of prime factors, HCF and LCM

Corbett MathsIndex laws and standard form

GCSE Maths TutorHighest Common Factor (HCF) | Higher & Foundation | GCSE Maths Tutor

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Key facts & methods

Prime factors, HCF and LCM

Product of prime factors: use a factor tree. Write answer using index notation. E.g. 72 = 2³ × 3².
HCF (Highest Common Factor): multiply shared prime factors. HCF(72, 60) → 72=2³×3², 60=2²×3×5 → HCF=2²×3=12.
LCM (Lowest Common Multiple): multiply each prime to its highest power. LCM(72,60)=2³×3²×5=360.
HCF is useful for simplifying fractions. LCM is useful for adding fractions and finding when events coincide.

Index laws

Multiply: aᵐ × aⁿ = aᵐ⁺ⁿ. E.g. 3⁴ × 3² = 3⁶.
Divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. E.g. 5⁶ ÷ 5² = 5⁴.
Power of a power: (aᵐ)ⁿ = aᵐⁿ. E.g. (2³)⁴ = 2¹².
Negative index: a⁻ⁿ = 1/aⁿ. E.g. 2⁻³ = 1/8.
Fractional index: a^(1/n) = ⁿ√a. E.g. 8^(1/3) = 2. Also: a^(m/n) = (ⁿ√a)ᵐ.
Zero index: a⁰ = 1 (for any non-zero a).

Standard form

Standard form: A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Large number: move decimal left → positive n. E.g. 450,000 = 4.5 × 10⁵.
Small number: move decimal right → negative n. E.g. 0.00037 = 3.7 × 10⁻⁴.
Multiply: multiply the A values, add the powers. (3 × 10⁴) × (2 × 10³) = 6 × 10⁷.
Divide: divide the A values, subtract the powers.
Add/subtract: convert to ordinary numbers first (or make powers equal).

Exam questions — 4 questions · 10 marks · Edexcel 4MA1 style

3 marksPrime factors + HCF4MA1 Set 1 Paper 1H style ✏️ No calculator

(a) Write 180 as a product of its prime factors. (b) Find the HCF of 180 and 126. Show your working. (3 marks)

Hint: (a) Factor tree for 180. (b) Factor tree for 126. HCF = product of shared prime factors, using the lower power of each.

+30 XP

3 marksIndex laws4MA1 style ✏️ No calculator

Simplify: (a) 5³ × 5⁴ ÷ 5⁵ (b) (3²)⁴ (c) Write 64^(−2/3) as a fraction. (3 marks)

Hint: (a) Add powers when multiplying, subtract when dividing. (b) Multiply the powers. (c) Fractional index: 1/3 means cube root. Negative means reciprocal.

+30 XP

4 marksStandard form calculations4MA1 Set 1 P1H style ✏️ No calculator

The distance from Earth to the Sun is approximately 1.5 × 10⁸ km. Light travels at 3 × 10⁵ km per second. Calculate the time, in seconds, for light to travel from the Sun to the Earth. Give your answer in standard form. (3 marks)

Hint: Time = distance ÷ speed. Divide the A values (1.5÷3) and subtract the powers (8−5). Adjust so A is between 1 and 10.

+40 XP

1 markNegative index4MA1 style

What is the value of 4⁻²?

A −16

B −8

C 1/16

D 1/8

Number, Indices & Standard Form

Prime factors, HCF and LCM

Index laws

Standard form

Module complete! 🎉