Mathematics • Year 9 • Unit 1 • Lesson 3
The Product Rule in the Real World
Use $a^m \times a^n = a^{m+n}$ in everyday contexts: stacking boxes, video uploads, file storage, school enrolment growth and biology lab cultures. Then explain your method in your own words.
1. Word problems
Each problem uses the product rule from Lesson 3: $a^m \times a^n = a^{m+n}$. Show your working — a single final answer with no working only earns half marks.
1.1 — Stacking storage boxes. Aaliyah packs her belongings into small boxes, then puts those into larger crates. There are $2^4$ small boxes per crate, and she has $2^3$ crates.
(a) Write the total number of small boxes as a product of two powers of $2$.
(b) Apply the product rule to express the total as a single power of $2$, then evaluate it. 3 marks
1.2 — Video uploads. A YouTuber uploads $3^2$ videos per month for $3^3$ months in a row.
(a) Write the total number of videos as a single power of $3$.
(b) Evaluate it as a normal number. 3 marks
1.3 — Phone storage breakdown. A phone has $2^{10}$ bytes per kilobyte and $2^{10}$ kilobytes per megabyte. So one megabyte equals $2^{10} \times 2^{10}$ bytes.
(a) Use the product rule to write one megabyte as a single power of $2$.
(b) Explain in one sentence what the index now represents. 3 marks
1.4 — School enrolment growth. A small school had $5^2$ students in $2020$. By $2024$, the number had grown by a factor of $5^3$.
(a) Write the new enrolment in $2024$ as a single power of $5$.
(b) Evaluate it as a normal number. (Hint: $5^5 = 3125$.) 3 marks
1.5 — Bacteria culture. A biology lab has a starting culture of $10^4$ bacteria. The culture multiplies by $10^2$ over the weekend, then multiplies by another $10$ on Monday morning.
(a) Write the final population as a product of three powers of $10$.
(b) Use the product rule to write it as a single power of $10$.
(c) Evaluate it 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 "$2^3 \times 3^4 = 6^7$". They're confident they're right. In your own words, explain (i) what mistake they have made, (ii) which condition of the product rule they have ignored, and (iii) the correct value of $2^3 \times 3^4$ as a normal number. Use the words "same base" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Stacking storage boxes
(a) Total $= 2^4 \times 2^3$ small boxes.
(b) Same base, add indices: $2^{4+3} = \mathbf{2^7} = 128$ small boxes.
Quick check: $2^4 = 16$ boxes per crate, $2^3 = 8$ crates, so $16 \times 8 = 128$ ✓.
1.2 — Video uploads
(a) Total $= 3^2 \times 3^3 = 3^{2+3} = \mathbf{3^5}$ videos.
(b) $3^5 = 243$ videos.
Quick check: $9 \times 27 = 243$ ✓.
1.3 — Phone storage breakdown
(a) $1$ MB $= 2^{10} \times 2^{10} = 2^{10+10} = \mathbf{2^{20}}$ bytes.
(b) The index $20$ counts the total number of $2$s multiplied — equivalently, the total number of "doublings" from $1$ byte to reach one megabyte.
So a megabyte ($2^{20}$ bytes) $= 1{,}048{,}576$ bytes — sometimes rounded to "about a million bytes".
1.4 — School enrolment growth
(a) New enrolment $= 5^2 \times 5^3 = 5^{2+3} = \mathbf{5^5}$ students.
(b) $5^5 = \mathbf{3125}$ students.
That's a 125-fold increase — clearly hypothetical for one school, but the index arithmetic is real!
1.5 — Bacteria culture
(a) Final population $= 10^4 \times 10^2 \times 10^1$.
(b) Same base $10$, add all three indices: $4 + 2 + 1 = 7$. So $\mathbf{10^7}$.
(c) $10^7 = \mathbf{10{,}000{,}000}$ bacteria (ten million).
For base $10$, the index = number of zeros.
2.1 — Explain your thinking (sample response)
My classmate has tried to use the product rule with two different bases ($2$ and $3$). The rule $a^m \times a^n = a^{m+n}$ only works when the bases are the same base. Adding the indices to get $6^7$ is meaningless here, because $2$ and $3$ are different numbers — you cannot combine them into base $6$. To work out $2^3 \times 3^4$ correctly, you have to evaluate each power separately: $2^3 = 8$ and $3^4 = 81$, then multiply normally: $8 \times 81 = \mathbf{648}$. A quick check: $6^7 = 279{,}936$, which is enormously larger than $648$ — the wrong answer is wildly off, confirming the rule was misused.
Marking: 1 mark for naming the rule's "same base" condition; 1 for identifying the bases are different; 1 for the correct answer $648$; 1 for a clear, full-sentence explanation.