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.

Answers, Do not peek before attempting

Section 2, We do (12y⁹ ÷ 4y³)

Step 1: (12 ÷ 4) × (y⁹ ÷ y ³).
Step 2: 12 ÷ 4 = 3.
Step 3: y⁹ ÷ y ³ = y^{(9 − 3)} = y ⁶.
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.