Mathematics • Year 10 • Unit 1 • Lesson 8
Index Laws in the Real World
Use Index Laws 1 and 2 (aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ) in everyday contexts: computer memory, bacteria growth, data file sizes, scientific notation for big and small numbers, and a band's streaming numbers.
1. Word problems
Each problem uses one of the index laws from Lesson 8: aᵐ × aⁿ = aᵐ⁺ⁿ or aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Show your working — final answers should be left as powers where the question asks.
1.1 — Computer memory (powers of 2). Computer memory is sold in powers of 2. A 4 GB stick is 2³² bytes. A 16 GB stick is 2³⁴ bytes.
(a) Write the total bytes of an installation with one 4 GB stick AND one 16 GB stick as a sum of two powers (you do not need to combine them — they have different "amounts" but the same base 2).
(b) Write 2³² × 2³⁴ as a single power of 2 (this would be the bytes if you had 4 GB per address of a 16 GB stick — not physically real, but a clean index-law calculation). 3 marks
1.2 — Bacteria doubling. A colony of E. coli doubles every hour. The starting number is N₀ = 2⁵ bacteria (32 cells). After h hours the population is N₀ × 2ʰ.
(a) Write the population after 7 hours as a single power of 2 (use Index Law 1).
(b) Evaluate that as a normal number. 2 marks
1.3 — Data file size ratio. A 4K video file is about 10¹⁰ bytes. A photo on the same camera is about 10⁷ bytes.
(a) Write the ratio video : photo as 10¹⁰ ÷ 10⁷ and simplify using Index Law 2.
(b) Convert your answer to a normal number — how many photos could fit in the space taken by one video? 2 marks
1.4 — Streaming numbers (band's monthly listeners). Indie band "Echo Hollow" has 3 × 10⁵ monthly Spotify listeners. A massive headliner has 2 × 10⁸ monthly listeners.
(a) Write the ratio (headliner ÷ Echo Hollow) using powers, then simplify.
(b) How many times larger is the headliner's audience? 3 marks
1.5 — Surface area scaling. The surface area of a sphere depends on the radius squared: S = 4πr². For a beachball with r = 30 cm, the surface area is 4π × 30² cm². For a soccer ball with r = 10 cm, the surface area is 4π × 10² cm².
(a) Write the ratio beachball : soccer ball as (4π × 30²) ÷ (4π × 10²) and simplify using Index Law 2 on the squares.
(b) How many times bigger is the beachball's surface area? 2 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "2³ × 5⁴ = 10⁷". They are confident the index laws let you "just add the indices and multiply the bases when both are different". In your own words, explain (i) why this is wrong, (ii) which condition for Index Law 1 is violated, and (iii) what the correct value of 2³ × 5⁴ actually is. Refer to "the bases must be the same" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Computer memory
(a) Total bytes = 2³² + 2³⁴. Note: this is a sum, not a product — Law 1 does NOT apply to sums.
(b) 2³² × 2³⁴ = 2³²⁺³⁴ = 2⁶⁶.
1.2 — Bacteria doubling
(a) Population after 7 hours = 2⁵ × 2⁷ = 2⁵⁺⁷ = 2¹².
(b) 2¹² = 4,096 bacteria.
1.3 — File size ratio
(a) Video / photo = 10¹⁰ ÷ 10⁷ = 10¹⁰⁻⁷ = 10³.
(b) 10³ = 1,000 photos fit in the same space as one 4K video.
1.4 — Spotify ratio
(a) (2 × 10⁸) ÷ (3 × 10⁵) = (2 ÷ 3) × (10⁸⁻⁵) = (2/3) × 10³.
(b) (2/3) × 1,000 ≈ 667 times larger.
1.5 — Beachball vs soccer ball surface area
(a) (4π × 30²) ÷ (4π × 10²) = 30² ÷ 10² = (30 ÷ 10)² = 3² = 9.
(Or: 30²/10² uses Index Law 2 on the squared parts directly — 900/100 = 9.)
(b) The beachball has 9 times the surface area of the soccer ball.
2.1 — Explain your thinking (sample response)
My classmate has broken the "the bases must be the same" condition for Index Law 1. The law aᵐ × aⁿ = aᵐ⁺ⁿ only works when both powers share the same base a. Here the bases are 2 and 5, which are different, so the law cannot be applied. There is no shortcut for 2³ × 5⁴ — we just calculate each part: 2³ = 8 and 5⁴ = 625, so 2³ × 5⁴ = 8 × 625 = 5,000. Note that 10⁷ = 10,000,000, which is two thousand times bigger than the real answer — a strong hint that the shortcut was wrong.
Marking: 1 mark for naming the condition; 1 for noting bases 2 and 5 are different; 1 for the correct value 5,000; 1 for a clear sentence-level explanation.