Mathematics • Year 9 • Unit 1 • Lesson 10

Mixed Index Laws in the Real World

Combine product, quotient, power, zero and negative index rules on numerical bases: storage drives, music apps, bracket sizes, savings accounts, and pixel grids. Then explain — in your own words — how to pick the right rule.

Apply · Real-World Maths

1. Word problems

Each problem mixes at least two of the five index rules from Lessons 1-10. Show your working; finish with a positive index.

1.1 — Storage capacity ratio. A laptop has $2^7$ GB of storage. A USB stick has $2^4$ GB; an old CD-R has $2^{-1}$ GB (i.e. half a GB). The ratio of laptop storage to (USB $\times$ CD-R) is $\dfrac{2^7}{2^4 \times 2^{-1}}$.

(a) Simplify the ratio to a single power of $2$.
(b) Evaluate it as a number, and write one sentence interpreting it (e.g. "the laptop is $\ldots$ times the combined storage"). 3 marks

Stuck? Simplify the bottom first: $2^4 \times 2^{-1} = 2^3$. Then quotient rule with the $2^7$ on top.

1.2 — Music quality halving. A music app lets you choose quality levels. Each lower level halves the bit-rate, so level $-2$ has bit-rate $2^{-2}$ of the original, and level $+4$ has bit-rate $2^4$. A "mixed playlist" averages a track at level $-2$ multiplied by a track at level $+4$: $2^{-2} \times 2^4$.

(a) Simplify $2^{-2} \times 2^4$ to a single power of $2$.
(b) Evaluate as a number — and check using the zero-index rule what $2^{-4} \times 2^4$ would give for comparison. 3 marks

Stuck? Product rule with negative indices: just add. Then negative-index rule if the answer is negative.

1.3 — Bracket expansion in pixel grids. A game tile is $3$ pixels wide. Stacking three of these tile-clusters together is $(3^3)^{-1}$ tile-cluster... wait, that gives a fractional grid! Let's instead consider $(3^{-2})^3 \times 3^4$ — a designer's note on rescaling.

(a) Simplify $(3^{-2})^3 \times 3^4$ to a single power of $3$.
(b) Evaluate as a fraction. Comment on whether the result is greater than $1$ or less than $1$. 3 marks

Stuck? Power-of-a-power FIRST (multiply $-2 \times 3$), then product rule (add).

1.4 — Compound interest snapshot. A savings account doubles its balance every year (a $\times 2$ growth factor). After $5$ years it would have grown by $2^5$; after $2$ years before today (i.e. in the past) it had only $2^{-2}$ of today's amount. The "growth from past to future" multiplier is $\dfrac{2^5}{2^{-2}}$.

(a) Simplify $\dfrac{2^5}{2^{-2}}$ to a single power of $2$ — be careful with "subtract a negative".
(b) Evaluate as a number; interpret what it means in plain English. 3 marks

Stuck on (a)? $2^{5-(-2)} = 2^{5+2}$ — subtract a negative becomes add.

1.5 — All five rules at once. A spreadsheet cell is $\dfrac{(2^3)^2 \times 2^{-4}}{2^0 \times 2^{-1}}$.

(a) Show every step in simplifying this to a single power of $2$.
(b) Evaluate as a number. 3 marks

Stuck? Power first ($(2^3)^2 = 2^6$), then top + bottom -, then watch for the $2^0 = 1$ and the subtract-a-negative on the bottom $2^{-1}$.

2. Explain your thinking

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

2.1 A classmate writes "$\dfrac{2^5 \times 2^{-3}}{2^4} = 2^{5 \times (-3) - 4} = 2^{-19}$, so the answer is $\dfrac{1}{2^{19}}$". In your own words, explain (i) what the right way of combining the indices in this expression is, (ii) where the classmate has confused multiplying with adding, and (iii) the correct final answer. Refer to "product rule" and "quotient rule" in your explanation.

Stuck? Revisit lesson § "Spot the Trap" — multiplying indices is ONLY for $(a^m)^n$, not for $a^m \times a^n$.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Storage capacity ratio

(a) Bottom: $2^4 \times 2^{-1} = 2^{4 + (-1)} = 2^3$. Whole: $\dfrac{2^7}{2^3} = 2^{7-3} = \mathbf{2^4}$.
(b) $2^4 = \mathbf{16}$. The laptop has $16$ times the combined USB-plus-CD storage.

1.2 — Music quality halving

(a) Product rule: $2^{-2 + 4} = \mathbf{2^2}$.
(b) $2^2 = \mathbf{4}$. For comparison: $2^{-4} \times 2^4 = 2^{-4+4} = 2^0 = 1$ — the lowest quality times the highest exactly cancels (zero-index rule).

1.3 — Bracket expansion

(a) Power: $(3^{-2})^3 = 3^{(-2) \times 3} = 3^{-6}$. Product: $3^{-6} \times 3^4 = 3^{-6+4} = \mathbf{3^{-2}}$.
(b) $3^{-2} = \dfrac{1}{3^2} = \mathbf{\dfrac{1}{9}}$. The result is LESS than $1$, because the negative index makes a fraction.

1.4 — Compound interest snapshot

(a) Quotient rule: $\dfrac{2^5}{2^{-2}} = 2^{5 - (-2)} = 2^{5+2} = \mathbf{2^7}$.
(b) $2^7 = \mathbf{128}$. From $2$ years before today to $5$ years after today (a total stretch of $7$ years of doubling), the balance grows by a factor of $128$.

1.5 — All five rules at once

(a) Power: $(2^3)^2 = 2^6$. Numerator: $2^6 \times 2^{-4} = 2^{6+(-4)} = 2^2$.
Denominator: $2^0 \times 2^{-1} = 1 \times 2^{-1} = 2^{-1}$ (zero-index rule plus the $2^{-1}$ stays).
Whole: $\dfrac{2^2}{2^{-1}} = 2^{2 - (-1)} = 2^{2+1} = \mathbf{2^3}$.
(b) $2^3 = \mathbf{8}$.

2.1 — Explain your thinking (sample response)

For an expression like $\dfrac{2^5 \times 2^{-3}}{2^4}$, all the bases are the same, so we should COLLECT the indices using the product rule on the top (ADD: $5 + (-3) = 2$) and the quotient rule across the fraction bar (SUBTRACT bottom from top: $2 - 4 = -2$). The correct overall index is $5 + (-3) - 4 = -2$.
The classmate has MULTIPLIED the top indices ($5 \times (-3) = -15$), which is wrong. Multiplying indices is only for the power-of-a-power rule, e.g. $(2^5)^{-3}$ — it does NOT apply to $2^5 \times 2^{-3}$. They've used the wrong rule.
Correct: $\dfrac{2^5 \times 2^{-3}}{2^4} = 2^{5 + (-3) - 4} = 2^{-2} = \mathbf{\dfrac{1}{4}}$.

Marking: 1 mark for naming the product rule (add); 1 for naming the quotient rule (subtract); 1 for identifying the "multiplied instead of added" error; 1 for the correct final answer $\tfrac{1}{4}$.