Introduction to Powers
A power consists of a base number and an exponent that indicates how many times to multiply the base by itself. For example, 2⁵ equals 2 × 2 × 2 × 2 × 2 = 32. Powers appear throughout Year 8 mathematics and form the foundation for understanding exponential growth, scientific notation, and algebraic expressions.
Why it matters
Powers model real-world exponential growth patterns across science, technology, and finance. Bacterial populations double every 20 minutes, meaning 1 bacterium becomes 2¹⁵ = 32,768 bacteria after 5 hours. Digital storage uses powers of 2 — a gigabyte contains 2³⁰ bytes, approximately 1 billion individual units. Financial compound interest follows power patterns: £1000 invested at 5% annual interest becomes £1000 × 1.05¹⁰ = £1628.89 after 10 years. Computing relies heavily on powers of 2, from processor speeds measured in gigahertz (10⁹ cycles per second) to internet data transmission rates. Understanding powers prepares learners for GCSE topics including indices laws, standard form notation for very large and small numbers, and quadratic equations where x² terms appear frequently.
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 8²?
Answer: 64
- Understand the notation → 8² = 8 × 8 — 8² means 8 multiplied by itself.
- Calculate → 8 × 8 = 64 — Multiply 8 by 8.
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 3125 as a power of 5
Answer: 5⁵
- Divide 3125 by 5 repeatedly → 3125 → 625 → 125 → 25 → 5 → 1 — Keep dividing by 5 until you reach 1. Count how many times.
- Count the divisions → 5 times — We divided 5 times, so 3125 = 5⁵.
Common mistakes
- Adding the base and exponent instead of multiplying: writing 3² = 5 instead of 3² = 9
- Confusing the base with the exponent: calculating 2³ as 3² = 9 instead of 2³ = 8
- Multiplying base by exponent: writing 4³ = 12 instead of 4³ = 64
- Forgetting that any number to power 0 equals 1: claiming 7⁰ = 0 instead of 7⁰ = 1