Mathematics • Year 9 • Unit 1 • Lesson 4
The Quotient Rule — Mixed Challenge
Pull together everything from Unit 1 so far: index notation (L1), evaluating powers including negatives and BIDMAS (L2), product rule (L3) and quotient rule (L4). Choose the right tool, spot a mistake, then take on an open-ended puzzle.
1. Mixed problems — pick the right rule
Each question uses a different combination of ideas from Lessons 1-4. Decide which rule applies before you start writing. Show your working. 3 marks each
1.1 Simplify $\;\dfrac{a^{12}}{a^7}$ and state the base and index of your answer.
1.2 Evaluate $\;\dfrac{2^{10}}{2^6}$ as a single power of $2$, then as a number.
1.3 Simplify $\;\dfrac{x^4 \times x^7}{x^5}$ in two steps: product rule on top, then quotient rule.
1.4 Simplify $\;\dfrac{18 k^9}{6 k^4}$ fully.
1.5 Simplify $\;\dfrac{y^3}{y^8}$, leaving your answer with a negative index. Then explain in one sentence what that negative index means.
1.6 Evaluate $\;\dfrac{(-3)^8}{(-3)^5}$. Use the quotient rule treating $-3$ as the base, then determine the sign of your final answer.
2. Find the mistake
Another student has tried to simplify $\;\dfrac{a^9}{a^3}$. 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 $\dfrac{a^9}{a^3}$:
Line 1: Same base $a$, so the quotient rule applies.
Line 2: Subtract indices: $3 - 9 = -6$.
Line 3: Base stays the same: $a$.
Line 4: So $\dfrac{a^9}{a^3} = a^{-6}$.
(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? The rule is top index MINUS bottom index. Which number is on top in this expression?3. Open-ended challenge — Three quotients that equal $3^4$
This question has more than one valid answer — there are several different quotients that work. 4 marks
3.1 Find three different quotients of the form $\dfrac{3^m}{3^n}$ — with $m$ and $n$ positive whole numbers, $m > n$, and at least two different values of $n$ across your three answers — that all equal $\;3^4$ (which is $81$).
For each quotient you find:
(i) Write it down.
(ii) Show the quotient-rule working that confirms it equals $3^4$.
(iii) State the rule you used.
Bonus: Your three quotients must not include $\dfrac{3^4}{3^0}$ (since $3^0 = 1$, that's trivial).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $a^{12} / a^7$
$a^{12-7} = \mathbf{a^5}$. Base $= a$, index $= 5$.
1.2 — $2^{10} / 2^6$
$2^{10-6} = \mathbf{2^4 = 16}$.
1.3 — $\dfrac{x^4 \times x^7}{x^5}$
Step 1 — product rule on top: $x^4 \times x^7 = x^{4+7} = x^{11}$.
Step 2 — quotient rule: $\dfrac{x^{11}}{x^5} = x^{11-5} = \mathbf{x^6}$.
1.4 — $18 k^9 / (6 k^4)$
Coefficients: $18 \div 6 = 3$. Indices on $k$: $9 - 4 = 5$. So $\mathbf{3 k^5}$.
1.5 — $y^3 / y^8$
$y^{3-8} = \mathbf{y^{-5}}$.
The negative index means "reciprocal" — equivalently $\dfrac{1}{y^5}$. We have more $y$s on the bottom than the top, so after cancelling, $5$ are left on the bottom.
1.6 — $(-3)^8 / (-3)^5$
Same base $-3$, quotient rule: $(-3)^{8-5} = (-3)^3$. Odd index keeps the sign: $(-3)^3 = -27$. Answer: $\mathbf{-27}$.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student has subtracted in the wrong order: they did "bottom minus top" ($3 - 9$) instead of "top minus bottom" ($9 - 3$). The quotient rule is $\dfrac{a^m}{a^n} = a^{m-n}$ — top index minus bottom, always in that order.
(c) Corrected working:
$\dfrac{a^9}{a^3}$ — same base $a$, quotient rule applies.
Subtract: top minus bottom: $9 - 3 = 6$.
Base stays the same: $a$.
So $\dfrac{a^9}{a^3} = \mathbf{a^6}$.
Quick verification by expansion: $9$ copies of $a$ on top, $3$ on bottom; $3$ cancel, leaving $6$ on top — $a^6$ ✓.
3 — Open-ended (sample solution)
We need $\dfrac{3^m}{3^n} = 3^4$, so $m - n = 4$ with $n \ge 1$ and $m > n$.
Quotient 1: $\dfrac{3^5}{3^1}$. Working: $3^{5-1} = 3^4$ ✓. Rule: quotient rule.
Quotient 2: $\dfrac{3^6}{3^2}$. Working: $3^{6-2} = 3^4$ ✓. Rule: quotient rule.
Quotient 3: $\dfrac{3^7}{3^3}$. Working: $3^{7-3} = 3^4$ ✓. Rule: quotient rule.
Other valid quotients: $\dfrac{3^8}{3^4}$, $\dfrac{3^{10}}{3^6}$, etc. — any pair with $m - n = 4$ and $n \ge 1$.
Marking: 1 mark per correct quotient with working — up to 3. 1 extra mark for clearly naming the quotient rule. Award full marks for any three distinct valid quotients other than $\dfrac{3^4}{3^0}$.