[Leejohnston]

Powers & Scientific Notation — Making Big & Small Numbers Easy

Powers (also called indices or exponents) are a quick way to write repeated multiplication. 
Scientific notation is a clean way to write very big or very small numbers.

These are essential skills for GCSE maths and all sciences.

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1. What Are Powers?

A power tells you how many times to multiply a number by itself.

Examples:
• 3² = 3 × 3 = 9 
• 5³ = 5 × 5 × 5 = 125 
• 10⁴ = 10,000 

Terminology: 
3² → 3 is the base, 2 is the index (or exponent).

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2. Key Power Rules

Rule 1: Add powers when multiplying same bases 
aᵐ × aⁿ = aᵐ⁺ⁿ 

Example: 
2³ × 2² = 2⁵ = 32

Rule 2: Subtract powers when dividing 
aᵐ ÷ aⁿ = aᵐ⁻ⁿ 

Example: 
10⁶ ÷ 10² = 10⁴

Rule 3: Power of a power 
(aᵐ)ⁿ = aᵐⁿ 

Example: 
(3²)³ = 3⁶ = 729

Rule 4: Negative powers 
a⁻ⁿ = 1 / aⁿ 

Example: 
10⁻³ = 1/1000 = 0.001

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3. What Is Scientific Notation?

Scientific notation writes numbers as:

(number between 1 and 10) × 10^(power)

Examples:
• 4,200 = 4.2 × 10³ 
• 0.005 = 5 × 10⁻³ 
• 73,000 = 7.3 × 10⁴ 

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4. Converting to Scientific Notation

Move the decimal point until you get a number between 1 and 10.

If you move left → positive power 
Example: 
52,000 → 5.2 × 10⁴

If you move right → negative power 
Example: 
0.0037 → 3.7 × 10⁻³

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5. Multiplying & Dividing in Scientific Notation

Multiply numbers, add powers 
(3 × 10²) × (2 × 10³) 
= 6 × 10⁵

Divide numbers, subtract powers 
(9 × 10⁶) ÷ (3 × 10²) 
= 3 × 10⁴

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6. Why This Matters

You’ll use powers and scientific notation in:
• GCSE Maths 
• GCSE Science 
• Chemistry (moles, atoms, small quantities) 
• Physics (distances, speeds, forces) 
• Computing & algorithms 
• Astronomy (huge distances) 

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Summary

Powers simplify repeated multiplication. 
Scientific notation simplifies very big or very small numbers.

Once you master these:
• Calculations become faster 
• GCSE questions become easier 
• Scientific data becomes clearer 

This skill is essential for higher-level work in maths and science.