Mathematics • Year 9 • Unit 1 • Lesson 6
Combining Index Laws in the Real World
Use the product, quotient and power-of-a-power rules together in everyday contexts: warehouse stacking, doubling-times, screen pixels, podcast downloads, and shrinking packaging. Then explain your method in your own words.
1. Word problems
Each problem combines two or three index laws from Lessons 3, 4 and 5. Show your working — a single final answer with no working only earns half marks.
1.1 — Warehouse stacking. A warehouse packs $2^3$ small boxes into each medium box, and $2^2$ medium boxes into each large crate. The team makes $2^4$ large crates per day, then ships $2^3$ days' worth of crates each week.
(a) Write the total number of small boxes shipped per week as a product of four powers of $2$.
(b) Apply the product rule to express it as a single power of $2$.
(c) Evaluate it as a number. 3 marks
1.2 — Doubling-time bacteria. A bacterial culture doubles every hour. After $h$ hours, the population is multiplied by $2^h$. Two cultures, A and B, both start with $1$ bacterium. Culture A grows for $3$ hours; Culture B grows for $5$ hours.
(a) Write the total population of A and B combined as $2^3 + 2^5$. Why can you NOT combine this as a single power of $2$?
(b) Now suppose instead the two cultures are mixed and grow for another $4$ hours together at the same doubling rate. Write the combined-then-grown size as $(2^3 + 2^5) \times 2^4$, and evaluate it. 3 marks
1.3 — Screen pixels. A phone screen is $2^{10}$ pixels wide and $2^{10}$ pixels tall, so it has $(2^{10})^2$ pixels in total. The phone's camera produces images that are $(2^{11})^2$ pixels.
(a) Use the power-of-a-power rule to express the screen size and the camera image size as single powers of $2$.
(b) Use the quotient rule to find how many times more pixels are in a camera image than on the screen. Express your answer as a single power of $2$. 3 marks
1.4 — Podcast downloads. A podcast had $(10^2)^3$ downloads in its first year. In its second year, it gained another $10^4$ downloads.
(a) Apply the power-of-a-power rule to express the first-year downloads as a single power of $10$.
(b) Use the product rule to write the total downloads (year 1 plus year 2 growth multiplied into year 1) as a single power of $10$. Specifically: in year 2 the count is multiplied by $10^4$. Total $=$ (year-1 count) $\times\ 10^4$. Evaluate the index. 3 marks
1.5 — Shrinking packaging. A product comes in a box of side $a$ cm, so its volume is $a^3$ cm³. The company tries two designs:
Design A: side length $a^2$ (much bigger).
Design B: half the original side length.
(a) Write Design A's volume using the power-of-a-power rule.
(b) Write the ratio "Design A's volume divided by the original $a^3$" as a single power of $a$, assuming $a > 1$. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "$(3^2)^3 \times 3^4 = 3^{2+3} \times 3^4 = 3^5 \times 3^4 = 3^9$". They've made one specific mistake but otherwise their working is fine. In your own words, explain (i) which rule they got wrong, (ii) why brackets need to be resolved first, and (iii) the correct simplification of $(3^2)^3 \times 3^4$. Use the words "power of a power" and "multiply" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Warehouse stacking
(a) Total $= 2^3 \times 2^2 \times 2^4 \times 2^3$ small boxes per week.
(b) Product rule (same base $2$): $2^{3+2+4+3} = \mathbf{2^{12}}$.
(c) $2^{12} = \mathbf{4096}$ small boxes per week.
Quick check: $8 \times 4 \times 16 \times 8 = 4096$ ✓.
1.2 — Doubling-time bacteria
(a) Total after the separate growth: $2^3 + 2^5 = 8 + 32 = 40$ bacteria. You can't write this as a single power of $2$ because $40$ is NOT a power of $2$. Index laws apply to multiplication and division, not addition.
(b) After mixing, then growing for $4$ more hours: $(2^3 + 2^5) \times 2^4 = 40 \times 16 = \mathbf{640}$ bacteria.
Common error: writing $2^3 + 2^5 = 2^8$ — that's wrong. $2^8 = 256$, not $40$.
1.3 — Screen pixels
(a) Screen: $(2^{10})^2 = 2^{20}$ pixels. Camera image: $(2^{11})^2 = 2^{22}$ pixels.
(b) Ratio $= \dfrac{2^{22}}{2^{20}} = 2^{22 - 20} = \mathbf{2^2 = 4}$ times more pixels in a camera image.
So a typical phone camera image has roughly $4$ times the pixel count of the screen — which is why zooming in still looks sharp.
1.4 — Podcast downloads
(a) $(10^2)^3 = 10^{2 \times 3} = \mathbf{10^6}$ downloads in year 1.
(b) Total $= 10^6 \times 10^4 = 10^{6 + 4} = \mathbf{10^{10}}$ downloads after the year-2 multiplication.
That's ten billion downloads — clearly hypothetical, but the index arithmetic is exact.
1.5 — Shrinking packaging
(a) Design A volume $= (a^2)^3 = a^{2 \times 3} = \mathbf{a^6}$ cm³.
(b) Ratio $= \dfrac{a^6}{a^3} = a^{6 - 3} = \mathbf{a^3}$.
For $a = 5$, that's a factor of $125$ bigger by volume — even a small change in side length makes a HUGE volume change.
2.1 — Explain your thinking (sample response)
My classmate has used the product rule (add indices) when they should have used the power of a power rule (multiply indices). The rule for brackets like $(3^2)^3$ is $(a^m)^n = a^{mn}$, which means MULTIPLY the inner and outer indices, not add them. So $(3^2)^3 = 3^{2 \times 3} = 3^6$, not $3^{2+3} = 3^5$. Brackets must be resolved first — before any multiplication outside — because the bracket is its own self-contained expression that the outer power applies to. The corrected working is: $(3^2)^3 \times 3^4 = 3^6 \times 3^4 = 3^{6 + 4} = \mathbf{3^{10}}$. (The classmate's $3^9$ answer is wrong; the correct answer is $3^{10}$.)
Marking: 1 mark for naming "power of a power" and "multiply"; 1 for explaining why brackets come first; 1 for the correct answer $3^{10}$; 1 for a clear, full-sentence explanation.