Mathematics • Year 9 • Unit 1 • Lesson 13

Quotient and Power Rules (Algebra)

Build fluency with $\dfrac{a x^m}{b x^n} = \dfrac{a}{b} x^{m-n}$ and $(a x^m)^n = a^n x^{mn}$. Coefficients use ARITHMETIC; indices use the LAW. Three sections: worked, guided, eight independent practice.

Build · I Do / We Do / You Do

1. I do — fully worked example

Watch what happens to the coefficient and to the variable separately. The coefficient is just division; the variable uses the quotient rule.

Problem. Simplify $\dfrac{12 x^4 y^3}{4 x y^2}$.

Step 1 — Divide the coefficients.

$\dfrac{12}{4} = 3$

Reason: coefficients use normal arithmetic — NOT the index rule.

Step 2 — Subtract the $x$ indices (top minus bottom).

$x^{4 - 1} = x^3$

Reason: quotient rule $\dfrac{x^m}{x^n} = x^{m-n}$. The lone $x$ on the bottom is $x^1$.

Step 3 — Subtract the $y$ indices.

$y^{3 - 2} = y^1 = y$

Reason: same rule, applied to the $y$ column.

Step 4 — Combine.

$\dfrac{12 x^4 y^3}{4 x y^2} = 3 x^3 y$

Reason: coefficient out the front, then each variable with its $m - n$ index.

Answer: $\mathbf{3 x^3 y}$.

Stuck? Revisit lesson § "Quotient rule with algebra". One column per variable; coefficient division stands alone.

2. We do — fill in the missing steps

Same structure as Section 1 but with the working faded. Fill in each blank. 4 marks

Problem. Expand $(2x^3)^4$.

Step 1 — Raise the coefficient to the outside power:

$2^4 = \_\_\_\_\_$

The $2$ is INSIDE the bracket too — it gets the power.

Step 2 — Multiply the variable's index by the outside power:

$x^{3 \times 4} = x^{\_\_\_}$

Step 3 — Combine:

$(2x^3)^4 = \_\_\_\_\_\_\_\_$

Stuck? Revisit lesson § "Power of a product". Every factor inside (including the $2$) gets the outer power.

3. You do — independent practice

Show working under each problem. Foundation = single rule. Standard = two rules combined. Extension = all three.

Foundation — single rule

3.1 Simplify $\dfrac{20 a^6}{5 a^2}$. 1 mark

3.2 Simplify $\dfrac{24 m^7}{6 m^3}$. 1 mark

3.3 Expand $(3 y^4)^2$. 1 mark

3.4 Expand $(4 b^2)^3$. 1 mark

Standard — combine two rules

3.5 Simplify $\dfrac{18 p^7 q^2}{6 p^3 q}$. 2 marks

3.6 Expand $(2 a b^2)^3$. 2 marks

Extension — combine all three

3.7 Simplify $\dfrac{(2 x^2)^3 \cdot x^4}{4 x^3}$. 3 marks

3.8 Simplify $\dfrac{(3 x^2)^2 \cdot x^3}{x^4}$. 2 marks

Stuck on 3.7? Order matters: apply the power FIRST ($(2x^2)^3 = 8 x^6$), then multiply in the numerator, then divide. Powers before products before quotients.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (faded $(2x^3)^4$)

Step 1: $2^4 = \mathbf{16}$.
Step 2: $x^{3 \times 4} = \mathbf{x^{12}}$.
Step 3: $(2x^3)^4 = \mathbf{16 x^{12}}$.

3.1 — $\dfrac{20 a^6}{5 a^2}$

$\dfrac{20}{5} \cdot a^{6-2} = \mathbf{4 a^4}$.

3.2 — $\dfrac{24 m^7}{6 m^3}$

$\dfrac{24}{6} \cdot m^{7-3} = \mathbf{4 m^4}$.

3.3 — $(3 y^4)^2$

$3^2 \cdot y^{4 \times 2} = \mathbf{9 y^8}$.

3.4 — $(4 b^2)^3$

$4^3 \cdot b^{2 \times 3} = \mathbf{64 b^6}$.

3.5 — $\dfrac{18 p^7 q^2}{6 p^3 q}$

$\dfrac{18}{6} \cdot p^{7-3} \cdot q^{2-1} = \mathbf{3 p^4 q}$.

3.6 — $(2 a b^2)^3$

Every factor inside gets the 3: $2^3 \cdot a^{1 \times 3} \cdot b^{2 \times 3} = \mathbf{8 a^3 b^6}$.

3.7 — $\dfrac{(2 x^2)^3 \cdot x^4}{4 x^3}$

Step 1 (power): $(2x^2)^3 = 8 x^6$.
Step 2 (product): $8 x^6 \cdot x^4 = 8 x^{10}$.
Step 3 (quotient): $\dfrac{8 x^{10}}{4 x^3} = 2 x^{10-3} = \mathbf{2 x^7}$.

3.8 — $\dfrac{(3 x^2)^2 \cdot x^3}{x^4}$

$(3x^2)^2 = 9 x^4$; numerator $9 x^4 \cdot x^3 = 9 x^7$; quotient $\dfrac{9 x^7}{x^4} = \mathbf{9 x^3}$.