Mathematics • Year 9 • Unit 1 • Lesson 11

Algebraic Index Laws in the Real World

Use the product, quotient and power rules on algebraic expressions in everyday contexts: rectangle areas, container volumes, speed and distance, plus the all-important "add or multiply?" decision.

Apply · Real-World Maths

1. Word problems

Each problem uses index laws on algebraic bases. Decide the OPERATION first ($+$ vs $\times$), then the RULE. Show your working.

1.1 — Rectangle area. A garden bed is a rectangle of length $4x^3$ metres and width $3x^2$ metres.

(a) Write an algebraic expression for the area in simplified form.
(b) Evaluate the area when $x = 2$. 3 marks

Stuck on (a)? Area = length $\times$ width. The operation is $\times$ — coefficients multiply, indices add.

1.2 — Adding pieces of fence. Sami is collecting fence segments. He has $6x^3$ metres on Monday and $4x^3$ metres on Tuesday — both with the same "$x^3$" length unit (the segments are all the same size).

(a) Write the TOTAL length of fence in simplified form.
(b) A classmate writes "$6x^3 + 4x^3 = 10 x^6$". State what's wrong and what the correct answer is. 3 marks

Stuck on (b)? Adding like terms keeps the index. Only multiplying adds indices.

1.3 — Speed and distance. A train moves at speed $20 m^8$ metres per second (where $m$ is a model parameter). It travels for $5 m^3$ seconds.

(a) Write the distance travelled in simplified form. (Distance = speed $\times$ time.)
(b) State which Lesson 11 rule you used. 3 marks

Stuck? Coefficients $20 \times 5 = 100$; same base $m$, so indices $8 + 3$.

1.4 — Cube volume revisited. A cube has side length $p^3$ cm.

(a) Write the volume of the cube in terms of $p$. (Volume = side $\times$ side $\times$ side, or $(\text{side})^3$.)
(b) State which Lesson 11 rule you used. Then evaluate the volume when $p = 2$. 3 marks

Stuck? Volume $= (p^3)^3$. Power-of-a-power: multiply the indices.

1.5 — Fractional fuel use. A car uses $\dfrac{20 m^8}{5 m^3}$ litres of fuel over a journey (some model formula).

(a) Simplify the fuel use expression.
(b) State which Lesson 11 rule you used. 3 marks

Stuck? Coefficients $20 \div 5$; same base $m$, indices $8 - 3$. That's the quotient rule on algebraic bases.

2. Explain your thinking

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

2.1 A classmate writes "$3x^2 \times 4x^2 = 7x^4$". In your own words, explain (i) what's wrong with the coefficient ($7$), (ii) the correct way to handle coefficients when multiplying terms, (iii) the correct final answer, and (iv) one similar example using addition where the coefficient WOULD be $7$ (so the student can see the difference).

Stuck? Revisit lesson § "Spot the Trap" — adding coefficients ($3 + 4 = 7$) is for ADDITION of like terms. Multiplying terms requires MULTIPLYING the coefficients.

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Answers — Do not peek before attempting

1.1 — Rectangle area

(a) Area $= 4x^3 \times 3x^2 = (4 \times 3) x^{3+2} = \mathbf{12 x^5}$ m$^2$.
(b) With $x = 2$: $12 \times 2^5 = 12 \times 32 = \mathbf{384}$ m$^2$.

1.2 — Adding pieces of fence

(a) Total $= 6x^3 + 4x^3 = (6 + 4)\,x^3 = \mathbf{10 x^3}$ metres. Like terms — index stays.
(b) The classmate has ADDED the indices ($3 + 3 = 6$) as if they were multiplying. Adding like terms keeps the index the same — only coefficients add. Correct answer: $\mathbf{10 x^3}$.

1.3 — Speed and distance

(a) Distance $= 20 m^8 \times 5 m^3 = (20 \times 5) m^{8+3} = \mathbf{100 m^{11}}$ metres.
(b) Rule used: product rule on the variable, with coefficients multiplied separately.

1.4 — Cube volume

(a) Volume $= (p^3)^3 = p^{3 \times 3} = \mathbf{p^9}$ cm$^3$.
(b) Rule used: power-of-a-power. With $p = 2$: $2^9 = \mathbf{512}$ cm$^3$.

1.5 — Fractional fuel use

(a) $\dfrac{20 m^8}{5 m^3} = \dfrac{20}{5} \times m^{8-3} = \mathbf{4 m^5}$ litres.
(b) Rule used: quotient rule on the variable, with coefficients divided separately.

2.1 — Explain your thinking (sample response)

The coefficient $7$ is wrong because the operation here is $\times$, not $+$. With $\times$, you have to MULTIPLY the coefficients ($3 \times 4 = 12$), not add them. The classmate has used the rule for adding like terms (where coefficients DO add).
The correct way to handle $3x^2 \times 4x^2$ is: (i) multiply the coefficients $3 \times 4 = 12$; (ii) apply the product rule to the variables $x^2 \times x^2 = x^{2+2} = x^4$; (iii) combine: $\mathbf{12 x^4}$.
Example where the coefficient IS $7$: $3x^2 + 4x^2 = 7x^2$. Here the operation is $+$ on LIKE TERMS, so we add coefficients ($3 + 4 = 7$) and keep the index the same. The two situations look similar but use opposite rules — the OPERATION decides everything.

Marking: 1 mark for naming the operation-based rule confusion; 1 for the correct treatment of coefficients in $\times$; 1 for the correct final answer $12x^4$; 1 for a valid addition example where $7$ legitimately appears.