Introduction to Powers
Year 8 students often struggle when first encountering powers notation like 3⁴ or 2⁵. Understanding that 4³ means 4 × 4 × 4 (not 4 × 3) forms the foundation for all subsequent work with indices in GCSE mathematics.
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Why it matters
Powers appear everywhere in real-world calculations. A football pitch's area of 100m × 70m requires understanding that 10² = 100. Mobile phone storage capacities use powers of 2: 64GB actually means 2⁶ × 10⁹ bytes. Population growth models rely on exponential functions where Britain's 67 million population might grow by 1.02³⁰ over 30 years. Engineering calculations use powers constantly—the strength of steel beams follows cubic relationships where doubling thickness increases strength by 2³ = 8 times. Even simple interest calculations use powers: £1000 at 5% compound interest becomes £1000 × 1.05¹⁰ after 10 years. GCSE students need solid power foundations before tackling surds, logarithms, and exponential graphs in Years 9-11.
How to solve introduction to powers
Powers — Introduction
- A power has a base and an exponent: 3⁴ means 3 × 3 × 3 × 3.
- Any number to the power 1 equals itself: a¹ = a.
- Any number to the power 0 equals 1: a⁰ = 1.
- Squaring (²) and cubing (³) are the most common powers.
Example: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
Worked examples
What is 5²?
Answer: 25
- Understand the notation → 5² = 5 × 5 — 5² means 5 multiplied by itself.
- Calculate → 5 × 5 = 25 — Multiply 5 by 5.
What is 2³?
Answer: 8
- Understand the notation → 2³ = 2 × 2 × 2 — 2³ means 2 multiplied by itself 3 times.
- Multiply step by step → 2 × 2 = 4 — First multiply 2 × 2.
- Multiply by base again → 4 × 2 = 8 — Then multiply the result by 2.
Write 243 as a power of 3
Answer: 3⁵
- Divide 243 by 3 repeatedly → 243 → 81 → 27 → 9 → 3 → 1 — Keep dividing by 3 until you reach 1. Count how many times.
- Count the divisions → 5 times — We divided 5 times, so 243 = 3⁵.
Common mistakes
- Students write 3⁴ = 12 instead of 81, multiplying base by exponent rather than using repeated multiplication: 3 × 3 × 3 × 3.
- Confusing 2⁰ = 0 instead of 2⁰ = 1, forgetting that any number to the power zero equals 1.
- Writing 5² + 5² = 5⁴ instead of 50, adding exponents when they should add the calculated values: 25 + 25.
- Calculating 4³ as 4 + 4 + 4 = 12 instead of 4 × 4 × 4 = 64, using addition rather than multiplication.