Mathematics • Year 10 • Unit 1 • Lesson 9
Powers in the Real World
Use the power-of-a-power, power-of-a-product and power-of-a-quotient rules in everyday contexts: stacks of boxes, scaling up a recipe, photo crops, gaming tournaments, and bacteria. Then explain your method.
1. Word problems
Each problem uses one (or more) of the rules from Lesson 9: (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, or (a/b)ⁿ = aⁿ/bⁿ. Show your working — a single final answer with no working only earns half marks.
1.1 — Cubes of cubes (storage room). A warehouse stacks 3² small cardboard boxes inside one medium box. It then stacks 3² medium boxes inside one large crate. Finally it stacks 3² large crates inside one shipping container.
(a) Write the total number of small boxes in one shipping container as a power of 3.
(b) Evaluate it as a normal number. 3 marks
1.2 — Scaling a brownie recipe. Maya's brownie recipe makes a square tray with side length a cm and uses an amount of batter proportional to a² (the area of the tray). Her cousin wants brownies in a tray twice as wide each way, so the new side length is 2a cm.
(a) Write the new tray's area in terms of a using power-of-a-product.
(b) How many times more batter does the bigger tray need? 3 marks
1.3 — Phone screen pixels. A square section of Sam's phone screen is 5³ pixels wide. The whole screen is 5² of those sections wide and 5² of those sections tall.
(a) Write the total width of the screen in pixels as a single power of 5 (use Index Law 1).
(b) Write the total area (width × height) as a single power of 5. 3 marks
1.4 — Gaming tournament rounds. A knockout esports tournament starts with 2⁷ players. After every round, exactly half are eliminated.
(a) Show that starting with 2⁷ players and halving 7 times leaves exactly 1 player using index laws.
(b) If a larger tournament has (2³)⁴ players, how many is that as a single power of 2? 3 marks
1.5 — Bacteria on your phone. A microbiology class measures that the bacteria on a clean phone screen grows by a factor of 4 every hour. After h hours there are N₀ × 4ʰ bacteria.
(a) Write the growth factor after h hours as a power of 2 (use power-of-a-power on 4 = 2²).
(b) By how many times does the count grow over 3 hours? Give as a power of 2 and as a normal number. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "(2x)³ = 2x³". They are confident they are right. In your own words, explain (i) what mistake they have made, (ii) which rule from Lesson 9 they have forgotten, and (iii) what the correct simplification is. Refer to "every factor inside the brackets" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cubes of cubes
(a) Each layer multiplies by 3², three times in total: total = 3² × 3² × 3² = (3²)³ = 3²ˣ³ = 3⁶.
(b) 3⁶ = 729 small boxes.
Why this is a power of a power: three stacked layers of 3² packed inside each other is exactly (3²)³, which equals 3⁶.
1.2 — Scaling a recipe
(a) New area = (2a)² = 2²a² = 4a² cm² (using power-of-a-product).
(b) The new tray needs 4 times the batter (4a² ÷ a² = 4). Doubling the side length doesn't double the batter — it quadruples it.
Real-world warning: this is why "doubling the recipe" by doubling the tray dimensions almost always makes too much batter.
1.3 — Phone screen pixels
(a) Width = 5² × 5³ = 5²⁺³ = 5⁵ pixels (Law 1 from Lesson 8).
(b) Area = (5⁵)² = 5⁵ˣ² = 5¹⁰ pixels² (power-of-a-power).
Check: 5¹⁰ = 9,765,625 — about 9.8 megapixels, realistic for a phone.
1.4 — Gaming tournament
(a) Players after 7 rounds = 2⁷ ÷ 2⁷ = 2⁷⁻⁷ = 2⁰ = 1 (Law 2 from Lesson 8).
(b) (2³)⁴ = 2³ˣ⁴ = 2¹² = 4,096 players.
Power of a power again — three indices inside, four outside, multiply to twelve.
1.5 — Bacteria
(a) 4ʰ = (2²)ʰ = 2²ʰ (power-of-a-power).
(b) Over 3 hours: 4³ = (2²)³ = 2²ˣ³ = 2⁶ = 64 times.
So a phone left for 3 hours has 64× the bacteria it started with — clean your phone!
2.1 — Explain your thinking (sample response)
My classmate has forgotten that every factor inside the brackets gets the outer power. They have only raised the x to the power of 3 and left the 2 untouched. The rule they have forgotten is the power-of-a-product rule: (ab)ⁿ = aⁿbⁿ. Applied to (2x)³, this gives 2³ × x³ = 8x³ — so the correct answer is 8x³, not 2x³. Quick check: try x = 1. Then (2 × 1)³ = 2³ = 8, but their answer gives 2 × 1³ = 2. The two clearly disagree.
Marking: 1 mark for naming the rule; 1 for "every factor"; 1 for the correct answer 8x³; 1 for a clear, full-sentence explanation.