Mathematics • Year 9 • Unit 1 • Lesson 20

Index Laws in the Real World

Use the full Unit 1 toolkit — product, quotient, power-of-a-power, zero, negative, fractional indices and scientific notation — in real contexts: red blood cells, doubling cells, currency conversion, photo cropping, and a battery percentage that halves each hour. Then explain which law you used and why.

Apply · Real-World Maths

1. Word problems

Each problem mixes ideas from several lessons in Unit 1. Pick the right index law(s) and show your working. Where the answer is a real-world quantity, give it in standard scientific notation to a sensible number of sig figs.

1.1 — Red blood cells. A red blood cell has a volume of $9.0 \times 10^{-14}$ L. An adult has about $5.0$ L of blood, of which $45\%$ is red blood cells.

(a) Find the total volume of red blood cells, in litres.
(b) Estimate the number of red blood cells, to 2 significant figures, in scientific notation. 3 marks

Stuck on (b)? Number = total RBC volume $\div$ volume of one cell. Coefficients $2.25 / 9.0$; indices $0 - (-14)$.

1.2 — Doubling cells. A single bacterium divides into two every hour. Starting with $1$ bacterium, after $h$ hours there are $2^h$ bacteria.

(a) How many bacteria after $10$ hours? Give your answer in scientific notation.
(b) How many after one full day ($24$ hours)? Give your answer in scientific notation, to 2 sig fig.
(c) Briefly: why do scientists need scientific notation for this kind of problem? 3 marks

Stuck on (b)? $2^{24}$ is a big number. Use a calculator with EE/EXP, or write $2^{24} = (2^{10})^{2.4}$ if you want a feel — but for the answer, just compute.

1.3 — Currency conversion in scientific notation. A small country has a national debt of $\$8.4 \times 10^{12}$. The exchange rate is $1$ AUD $= 7.0 \times 10^{-1}$ USD.

(a) Convert the debt to AUD by dividing — give the answer in standard scientific notation.
(b) State the order of magnitude of the debt in AUD. 3 marks

Stuck on (a)? $\dfrac{\$ \text{USD}}{\text{USD per AUD}} = \text{AUD}$. Divide coefficients, subtract indices, re-standardise.

1.4 — Photo cropping (powers of a product). A square photo is $4 \times 10^3$ pixels on a side.

(a) Use $\text{Area} = \text{side}^2$ to find the total pixel area, in scientific notation. (Hint: use power-of-a-product on $(4 \times 10^3)^2$.)
(b) The photo is then re-cropped to half the width and half the height. What's the new area, in scientific notation, and what fraction of the original is it? 3 marks

Stuck on (a)? $(4 \times 10^3)^2 = 4^2 \times (10^3)^2 = 16 \times 10^6$, then re-standardise.

1.5 — Battery percentage (negative indices). A faulty phone battery loses half its charge each hour. Starting at full charge ($100\%$), the percentage after $h$ hours is $100 \times \left(\dfrac{1}{2}\right)^h = 100 \times 2^{-h}$.

(a) Use the negative-index rule to rewrite $2^{-h}$ as a positive-index expression.
(b) What percentage remains after $5$ hours? (Use $2^5 = 32$.)
(c) After how many hours does the battery first drop below $1\%$? 3 marks

Stuck on (c)? $100 / 2^h < 1$ means $2^h > 100$. List: $2^6 = 64$, $2^7 = 128$. So $h = 7$ is the first hour.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate is simplifying $\dfrac{(2 a^3)^2 \cdot a^{-1}}{4 a^{1/2}}$ and asks you: "How do I know which law to use first?" In your own words, write a short guide for them: (i) what's the right order to attack a mixed expression like this, (ii) why "brackets first" matters, (iii) which two laws are most often confused (add vs multiply on indices), and (iv) what final-check question you should always ask before writing the answer down (about positive indices, or the form of scientific notation).

Stuck? Revisit lesson § "Top Mistakes To Avoid In The Exam" and § "A four-law expression done slowly" — the guidance is right there.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Red blood cells

(a) RBC volume $= 5.0 \times 0.45 = \mathbf{2.25}$ L.
(b) Number $= \dfrac{2.25}{9.0 \times 10^{-14}} = \dfrac{2.25}{9.0} \times 10^{0 - (-14)} = 0.25 \times 10^{14} = \mathbf{2.5 \times 10^{13}}$ red blood cells (2 s.f.).
That's about 25 trillion red blood cells per adult — and your body replaces them every few months.

1.2 — Doubling cells

(a) After $10$ h: $2^{10} = 1024 \approx \mathbf{1.0 \times 10^3}$ bacteria.
(b) After $24$ h: $2^{24} = 16{,}777{,}216 \approx \mathbf{1.7 \times 10^7}$ bacteria (2 s.f.).
(c) Scientific notation is needed because exponential doubling makes numbers grow huge very quickly — by day 2 or 3, you'd be writing dozens of digits. Standard form keeps the size visible (the index) and the precision controllable (the coefficient).

1.3 — Currency conversion

(a) AUD debt $= \dfrac{8.4 \times 10^{12}}{7.0 \times 10^{-1}} = \dfrac{8.4}{7.0} \times 10^{12 - (-1)} = 1.2 \times 10^{13}$ AUD. Answer: $\mathbf{1.2 \times 10^{13}}$ AUD.
(b) Order of magnitude $\approx \mathbf{10^{13}}$ (tens of trillions).

1.4 — Photo cropping

(a) Area $= (4 \times 10^3)^2 = 4^2 \times (10^3)^2 = 16 \times 10^6 = \mathbf{1.6 \times 10^7}$ pixels$^2$ (power-of-a-product, then re-standardise).
(b) New side $= 2 \times 10^3$ pixels (half). New area $= (2 \times 10^3)^2 = 4 \times 10^6 = \mathbf{4 \times 10^6}$ pixels$^2$. Fraction of original $= \dfrac{4 \times 10^6}{1.6 \times 10^7} = 0.25 = \mathbf{\dfrac{1}{4}}$.
Halving both side lengths quarters the area — same as the L05 brownie scaling pattern.

1.5 — Battery percentage

(a) $2^{-h} = \dfrac{1}{2^h}$ (negative-index rule). So $100 \times 2^{-h} = \dfrac{100}{2^h}$.
(b) After $5$ h: $\dfrac{100}{2^5} = \dfrac{100}{32} = \mathbf{3.125\%}$ (about $3.1\%$).
(c) We need $\dfrac{100}{2^h} < 1$, i.e. $2^h > 100$. From $2^6 = 64 < 100$ and $2^7 = 128 > 100$, the answer is $h = \mathbf{7}$ hours.

2.1 — Explain your thinking (sample guide)

A short guide for a mixed index-law expression: (i) Order: brackets first, then $\times$, then $\div$. Apply power-of-a-product to anything in brackets before you start multiplying or dividing the whole expression. (ii) Brackets first matters because the outer power applies to every factor inside — including the coefficient. If you skip the bracket, you'll forget to power the coefficient (e.g. $(2 a)^3 = 8 a^3$, not $2 a^3$). (iii) Two laws are most often confused: the product rule (add indices: $x^m \cdot x^n = x^{m+n}$) and power-of-a-power (multiply indices: $(x^m)^n = x^{mn}$). Multiplying bases means adding indices; only the power-of-a-power law multiplies them. (iv) Final-check question: "Did the question ask for a positive index? If so, are there any negatives left to flip? If it's scientific notation, is the coefficient in $[1, 10)$?" These last checks catch most "right value, wrong form" errors.

Marking: 1 mark for the correct order; 1 for "every factor inside the brackets"; 1 for naming add vs multiply confusion; 1 for stating the final-check question.