Mathematics • Year 9 • Unit 1 • Lesson 11
Index Laws — Mixed Challenge
Pull together every index rule from Unit 1 (L1-L11) on both numerical and algebraic bases. Spot a mistake, decide between add and multiply, and design your own combined-operation expression.
1. Mixed problems — choose the right rule
Each question uses a different combination of rules from Lessons 1-11. Decide the OPERATION and the RULE before you start writing. Show your working. 3 marks each
1.1 Simplify $6x^3 + 4x^3$.
1.2 Simplify $6x^3 \times 4x^2$.
1.3 Simplify $\dfrac{20 m^8}{5 m^3}$.
1.4 Simplify $(2x^3)^2 \times 3x$.
1.5 Simplify $\dfrac{(2x^2)^3 \times (3x)^2}{6x^4}$ fully.
1.6 Simplify $2x^3 + 3x^3 + 2x^3 \times 3x^2$. Remember $\times$ binds before $+$.
2. Find the mistake
Another student has tried to simplify $2x^2 + 5x^2$. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — simplify $2x^2 + 5x^2$:
Line 1: $2x^2 + 5x^2$ — same base $x$, same index $2$, so we can combine.
Line 2: Coefficients $2 + 5 = 7$. ✓
Line 3: Indices $2 + 2 = 4$. ✓
Line 4: So $2x^2 + 5x^2 = 7 x^4$.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Indices only ADD when the operation is $\times$. Here the operation is $+$ on like terms, so the index STAYS THE SAME.3. Open-ended challenge — design an expression that hides a $+$ inside a $\times$
This question has more than one valid answer. 4 marks
3.1 Design two different algebraic expressions that each simplify to $\mathbf{10 x^5}$. Rules: each expression must use at least TWO of the index laws (product / quotient / power), and the two expressions must not be identical. At least one of them must mix an ADDITION of like terms with a MULTIPLICATION elsewhere (so a student would have to choose the right operation in each spot).
For each expression you build:
(i) Write it down clearly.
(ii) Show step-by-step working that confirms it equals $10 x^5$.
(iii) Name the index law(s) you used at each step.
Hint to start: $10x^5$ has coefficient $10$ and index $5$. So you might use $2 \times 5 = 10$ and $2 + 3 = 5$, e.g. $2x^2 \times 5x^3$. Or $4x^5 + 6x^5 = 10x^5$ for the addition route.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $6x^3 + 4x^3$
Like terms (same base, same index): $(6 + 4) x^3 = \mathbf{10 x^3}$. Index stays.
1.2 — $6x^3 \times 4x^2$
Coefficients $6 \times 4 = 24$; indices $3 + 2 = 5$. Answer: $\mathbf{24 x^5}$.
1.3 — $\dfrac{20 m^8}{5 m^3}$
Coefficients $20 \div 5 = 4$; indices $8 - 3 = 5$. Answer: $\mathbf{4 m^5}$.
1.4 — $(2x^3)^2 \times 3x$
Power-of-a-product: $(2x^3)^2 = 2^2 (x^3)^2 = 4 x^6$.
Multiply by $3x$: $4 x^6 \times 3 x = (4 \times 3) x^{6+1} = \mathbf{12 x^7}$.
1.5 — $\dfrac{(2x^2)^3 \times (3x)^2}{6x^4}$
Step 1 — power-of-a-product on each bracket:
$(2x^2)^3 = 2^3 (x^2)^3 = 8 x^6$.
$(3x)^2 = 3^2 x^2 = 9 x^2$.
Step 2 — multiply the numerator: $8 x^6 \times 9 x^2 = 72 x^{6+2} = 72 x^8$.
Step 3 — divide by $6 x^4$: $\dfrac{72 x^8}{6 x^4} = 12 \times x^{8-4} = \mathbf{12 x^4}$.
1.6 — $2x^3 + 3x^3 + 2x^3 \times 3x^2$
Multiplication first ($\times$ binds before $+$): $2x^3 \times 3x^2 = 6 x^{3+2} = 6 x^5$.
Now: $2x^3 + 3x^3 + 6x^5$. The first two are like terms: $2x^3 + 3x^3 = 5 x^3$.
Final: $\mathbf{5 x^3 + 6 x^5}$. (These two terms have different indices, so they cannot combine further.)
2 — Find the mistake
(a) The mistake is on Line 3 (with the wrong final answer on Line 4).
(b) Indices only ADD when the operation between the terms is $\times$. Here the operation is $+$ — so the index STAYS THE SAME (no addition of indices). The student has confused like-term addition with the product rule.
(c) Corrected working:
Line 1: $2x^2 + 5x^2$ — like terms.
Line 2: Coefficients $2 + 5 = 7$.
Line 3: Index stays $2$ (NOT $2 + 2 = 4$).
Line 4: So $2x^2 + 5x^2 = \mathbf{7 x^2}$.
This is the exact trap flagged in the lesson's "Spot the Trap" card — and one of the most common mistakes in Year 9 algebra.
3 — Open-ended challenge (sample solutions)
Expression 1 (pure product rule): $2x^2 \times 5x^3$.
Working: Coefficients $2 \times 5 = 10$; indices $2 + 3 = 5$. So $2x^2 \times 5x^3 = 10 x^5$ ✓.
Rule used: product rule with coefficients.
Expression 2 (addition mixed with multiplication): $4x^5 + 6x^5$ — but that's just addition. A better mix: $4x^5 + (2x \times 3x^4)$.
Working: Multiplication first: $2x \times 3x^4 = 6 x^{1+4} = 6 x^5$. Then like-term addition: $4x^5 + 6 x^5 = (4 + 6) x^5 = 10 x^5$ ✓.
Rules used: product rule (for the multiplication), then like-term addition (index stays).
Other valid approaches: $\dfrac{50 x^7}{5 x^2}$ uses the quotient rule (coefficients $50 \div 5 = 10$, indices $7 - 2 = 5$); $(2x)^2 \times \dfrac{5 x^3}{2}$ uses a bracket plus a fraction. Award full marks for any two valid, distinct expressions that meet the brief.
Marking: 2 marks per valid expression (one for the expression itself with named rules, one for clear step-by-step working that lands at $10 x^5$). Up to 4 in total. At least ONE of the two must include an addition-of-like-terms step alongside a multiplication.