Mathematics • Year 9 • Unit 1 • Lesson 4
The Quotient Rule
Build fluency with $\dfrac{a^m}{a^n} = a^{m-n}$: when dividing powers with the SAME base, SUBTRACT the indices (top minus bottom). One step at a time, from a fully worked example through guided practice to independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason on the right.
Problem. Simplify $\;\dfrac{7^8}{7^3}$. Leave your answer in index form.
Step 1 — Check the bases.
Both bases are $7$.
Reason: the quotient rule $\dfrac{a^m}{a^n} = a^{m-n}$ only applies when the bases are the SAME.
Step 2 — Subtract: top index MINUS bottom index.
$8 - 3 = 5$
Reason: $8$ copies of $7$ on top, $3$ cancel against the bottom, leaving $8 - 3 = 5$ copies on top.
Step 3 — Write the result with the same base.
$\dfrac{7^8}{7^3} = 7^5$
Reason: the base $7$ does NOT change — only the index changes.
Step 4 — (Optional) verify by expanding.
$\dfrac{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7} = 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 7^5$ ✓
Answer: $\mathbf{7^5}$ (which equals $16{,}807$).
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Simplify $\;\dfrac{x^9}{x^4}$.
Step 1 — Check the bases: both bases are __________ .
Step 2 — Subtract (top minus bottom):
$9 - 4 = \_\_\_$
Step 3 — Write the result with the same base:
$\dfrac{x^9}{x^4} = x^{\_\_\_}$
Step 4 — Verify by expanding: there are $\_\_$ copies of $x$ on top and $\_\_$ on the bottom; cancelling leaves $\_\_$ copies on top.
Answer: $\mathbf{x^{\_\_}}$.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single rule). The middle two are standard (combine two ideas). The last two are extension (push your thinking).
Foundation — single rule
3.1 Simplify $\;\dfrac{5^7}{5^2}$. 1 mark
3.2 Simplify $\;\dfrac{a^{10}}{a^6}$. 1 mark
3.3 Simplify $\;\dfrac{6^5}{6}$. (Don't forget $6 = 6^1$.) 1 mark
3.4 Evaluate $\;\dfrac{3^7}{3^4}$ as a single power of $3$, then as a number. 1 mark
Standard — combine two ideas
3.5 Simplify $\;\dfrac{y^4}{y^9}$, leaving your answer with a negative index. 2 marks
3.6 Simplify $\;\dfrac{12 m^7}{4 m^3}$ fully (handle coefficients and indices separately). 2 marks
Extension — push your thinking
3.7 Simplify $\;\dfrac{a^5 \times a^4}{a^6}$ in two steps: apply the product rule on top, then the quotient rule. 3 marks
3.8 Find $n$ if $\;\dfrac{x^{12}}{x^n} = x^4$. Explain how you used the quotient rule. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $x^9 / x^4$)
Step 1: both bases are $x$.
Step 2: $9 - 4 = \mathbf{5}$.
Step 3: $\dfrac{x^9}{x^4} = \mathbf{x^5}$.
Step 4: $\mathbf{9}$ copies of $x$ on top and $\mathbf{4}$ on the bottom; cancelling leaves $\mathbf{5}$ copies on top.
Answer: $\mathbf{x^5}$.
3.1 — $5^7 / 5^2$
$5^{7-2} = \mathbf{5^5}$.
3.2 — $a^{10} / a^6$
$a^{10-6} = \mathbf{a^4}$.
3.3 — $6^5 / 6$
$6 = 6^1$, so $6^{5-1} = \mathbf{6^4}$.
3.4 — $3^7 / 3^4$
$3^{7-4} = \mathbf{3^3 = 27}$.
3.5 — $y^4 / y^9$
Top minus bottom: $4 - 9 = -5$. So $\dfrac{y^4}{y^9} = \mathbf{y^{-5}}$.
Equivalently $\dfrac{1}{y^5}$ — preview of negative indices in Lesson 8.
3.6 — $12m^7 / (4m^3)$
Coefficients: $\dfrac{12}{4} = 3$. Indices on $m$: $7 - 3 = 4$. So $\mathbf{3 m^4}$.
3.7 — $\dfrac{a^5 \times a^4}{a^6}$
Step 1 — product rule on top: $a^5 \times a^4 = a^{5+4} = a^9$.
Step 2 — quotient rule: $\dfrac{a^9}{a^6} = a^{9-6} = \mathbf{a^3}$.
3.8 — Solve $x^{12} / x^n = x^4$
By the quotient rule, $x^{12 - n} = x^4$. Match indices: $12 - n = 4$, so $\mathbf{n = 8}$.
The rule says we subtract the bottom index from the top, so the resulting index is $12 - n$. Set that equal to $4$ and solve.