Mathematics • Year 9 • Unit 1 • Lesson 5
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 on your phone. Then explain your method in your own words.
1. Word problems
Each problem uses one (or more) of the rules from Lesson 5: $(a^m)^n = a^{mn}$, $(ab)^n = a^n b^n$, or $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$. Show your working — a single final answer with no working only earns half marks.
1.1 — Cubes of cubes (storage room). A warehouse stacks $3^2$ small cardboard boxes inside one medium box. It then stacks $3^2$ medium boxes inside one large crate. Finally it stacks $3^2$ 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 recipe. Maya's brownie recipe makes a square tray of brownies with side length $a$ cm and uses an amount of batter proportional to $a^2$ (the area of the tray). Her cousin's family 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 compared with the original? 3 marks
1.3 — Phone screen pixels. A square section of Sam's phone screen is $5^3$ pixels wide. The whole screen is $5^2$ of those sections wide and $5^2$ of those sections tall.
(a) Write the total width of the screen in pixels as a single power of $5$.
(b) Write the total area of the screen (width $\times$ height) as a single power of $5$. 3 marks
1.4 — Gaming tournament rounds. A knockout esports tournament starts with $2^7$ players. After every round, exactly half of the remaining players are eliminated. The organisers describe the bracket as "$7$ rounds of halving — that's $(2)^7$ players knocked out in total".
(a) Show that starting with $2^7$ players and halving $7$ times leaves exactly $1$ player using index laws.
(b) If a larger tournament instead has $(2^3)^4$ players, how many players is that, written as a single power of $2$? 3 marks
1.5 — Bacteria on your phone. A microbiology class measures that the number of bacteria on a clean phone screen grows by a factor of $4$ every hour. After $h$ hours there are $N_0 \times 4^h$ bacteria, where $N_0$ is the starting count.
(a) Write the growth factor after $h$ hours as a power of $2$ (use power-of-a-power on $4 = 2^2$).
(b) By how many times does the bacteria count grow over $3$ hours? Give your answer 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 "$(2x)^3 = 2x^3$". They're confident they're right. In your own words, explain (i) what mistake they have made, (ii) which rule from Lesson 5 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 $3^2$, three times in total: total $= 3^2 \times 3^2 \times 3^2 = (3^2)^3 = 3^{2 \times 3} = \mathbf{3^6}$.
(b) $3^6 = 729$ small boxes.
Why this is a power of a power: three stacked layers of $3^2$ packed inside each other is exactly $(3^2)^3$, which by the rule equals $3^6$.
1.2 — Scaling a recipe
(a) New area $= (2a)^2 = 2^2 a^2 = \mathbf{4 a^2}$ cm$^2$ (using power-of-a-product).
(b) The new tray needs $\mathbf{4}$ times the batter — because $\dfrac{4a^2}{a^2} = 4$. 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 $= 5^2 \times 5^3 = 5^{2+3} = \mathbf{5^5}$ pixels (product rule, from L3).
(b) Area $= (5^5)^2 = 5^{5 \times 2} = \mathbf{5^{10}}$ pixels$^2$ (power-of-a-power).
If you want a check: $5^{10} = 9{,}765{,}625$ — about 9.8 megapixels, which is realistic for a phone.
1.4 — Gaming tournament rounds
(a) Players after $7$ rounds $= \dfrac{2^7}{2^7} = 2^{7-7} = 2^0 = \mathbf{1}$. (Uses the quotient rule from L4.)
(b) $(2^3)^4 = 2^{3 \times 4} = \mathbf{2^{12}} = 4096$ players.
Power of a power again — three indices inside, four outside, multiply to get twelve.
1.5 — Bacteria on your phone
(a) $4^h = (2^2)^h = \mathbf{2^{2h}}$ (power-of-a-power).
(b) Over $3$ hours: $4^3 = (2^2)^3 = 2^{2 \times 3} = \mathbf{2^6 = 64}$ times.
So a phone left for $3$ hours has $64$ times 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: $(ab)^n = a^n b^n$. Applied to $(2x)^3$, this gives $2^3 \times x^3$, and $2^3 = 8$, so the correct answer is $\mathbf{8x^3}$ — not $2x^3$. A quick check: try $x = 1$. Then $(2 \times 1)^3 = 2^3 = 8$, but their answer gives $2 \times 1^3 = 2$. The two clearly disagree, which confirms the mistake.
Marking: 1 mark for naming the rule; 1 for "every factor"; 1 for the correct answer $8x^3$; 1 for a clear, full-sentence explanation.