Mathematics • Year 10 • Unit 1 • Lesson 8

Index Laws — Multiplication and Division

Build fluency with the two basic index laws from Lesson 8: aᵐ × aⁿ = aᵐ⁺ⁿ (same base, add indices) and aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, subtract indices). One step at a time, from a worked example through guided practice to independent problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every line. Each step has a short reason so you can see why, not just what.

Problem. Simplify 2x³ × 5x⁴. Leave your answer in index form.

Step 1 — Separate numbers from pronumerals.

2x³ × 5x⁴ = (2 × 5) × (x³ × x⁴)

Reason: multiplication is commutative — we can rearrange to group like things together.

Step 2 — Multiply the numerical coefficients.

2 × 5 = 10

Reason: just ordinary multiplication. Indices only apply to bases.

Step 3 — Apply Index Law 1 to the same-base powers.

x³ × x⁴ = x³⁺⁴ = x⁷

Reason: aᵐ × aⁿ = aᵐ⁺ⁿ — same base x, so ADD the indices.

Step 4 — Put it together.

2x³ × 5x⁴ = 10x⁷

Reason: write the coefficient out the front, then the simplified pronumeral.

Answer: 10x⁷.

Stuck? Revisit lesson § "Multiplying Powers with the Same Base" — the expanded form there shows why we add the indices.

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. Simplify (12y⁹) ÷ (4y³).

Step 1 — Split the coefficients and the pronumerals:

(12y⁹) ÷ (4y³) = (12 ÷ ____) × (y⁹ ÷ y____)

Step 2 — Divide the coefficients:

12 ÷ ____ = ______

Step 3 — Apply Index Law 2 (same base, subtract):

y⁹ ÷ y____ = y^(9 − ____) = y____

Step 4 — Put it together:

(12y⁹) ÷ (4y³) = __________

Stuck? Revisit lesson § "Worked Example 4 — Division with Algebraic Bases" for the same pattern.

3. You do — independent practice

Show your working. The first four are foundation (single law). The middle two are standard (combine numbers and pronumerals). The last two are extension (multiple terms).

Foundation — single rule

3.1 Simplify 5⁶ × 5³. Leave your answer as a power of 5. 1 mark

3.2 Simplify x⁴ × x⁷. 1 mark

3.3 Simplify 7⁹ ÷ 7⁴. Leave as a power of 7. 1 mark

3.4 Simplify y¹² ÷ y⁵. 1 mark

Standard — combine numbers and pronumerals

3.5 Simplify 3a⁵ × 4a². 2 marks

3.6 Simplify (20m⁸) ÷ (5m³). 2 marks

Extension — push your thinking

3.7 Simplify (2x³ × 6x⁵) ÷ (4x²). 3 marks

3.8 Find the value of n in x⁸ × xⁿ = x¹⁵. Explain how you used Index Law 1. 2 marks

Stuck on 3.7? Simplify the numerator first using Law 1, then divide using Law 2.

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Section 2 — We do (12y⁹ ÷ 4y³)

Step 1: (12 ÷ 4) × (y⁹ ÷ y³).
Step 2: 12 ÷ 4 = 3.
Step 3: y⁹ ÷ y³ = y^{(9 − 3)} = y⁶.
Step 4: (12y⁹) ÷ (4y³) = 3y⁶.

3.1 — 5⁶ × 5³

aᵐ × aⁿ = aᵐ⁺ⁿ, so 5⁶ × 5³ = 5⁶⁺³ = 5⁹.

3.2 — x⁴ × x⁷

x⁴⁺⁷ = x¹¹.

3.3 — 7⁹ ÷ 7⁴

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, so 7⁹⁻⁴ = 7⁵.

3.4 — y¹² ÷ y⁵

y¹²⁻⁵ = y⁷.

3.5 — 3a⁵ × 4a²

(3 × 4) × (a⁵ × a²) = 12 × a⁵⁺² = 12a⁷.

3.6 — 20m⁸ ÷ 5m³

(20 ÷ 5) × (m⁸ ÷ m³) = 4 × m⁸⁻³ = 4m⁵.

3.7 — (2x³ × 6x⁵) ÷ (4x²)

Numerator: 2x³ × 6x⁵ = 12 × x³⁺⁵ = 12x⁸.
Divide: 12x⁸ ÷ 4x² = 3 × x⁸⁻² = 3x⁶.

3.8 — Solve x⁸ × xⁿ = x¹⁵

By Index Law 1, x⁸ × xⁿ = x⁸⁺ⁿ. Match indices: 8 + n = 15, so n = 7.
The law says we add the indices. The combined index 8 + n must equal 15, so we solve for n.