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Stage 4 · Indices · MA4-IND-C-01

Index Laws with Positive Indices

Think first: Can you evaluate 2⁷ × 2⁵ without writing twelve factors of 2?

Open practice worksheet

Know

  • product, quotient and power-of-a-power laws
  • the same-base condition
  • the square-of-a-product relationship

Understand

  • each law follows by counting repeated factors
  • why (ab)² = a²b²
  • why indices cannot be combined when bases differ

Can do

  • establish laws from expanded form
  • verify numerical product powers
  • simplify numerical expressions with positive indices
1

Multiplying same bases

Joining repeated factors adds their counts:

3² × 3⁴ = (3 × 3)(3 × 3 × 3 × 3) = 3⁶

So aᵐ × aⁿ = aᵐ⁺ⁿ for the same non-zero base.

2

Dividing same bases

Cancel matching factors:

5⁶ ÷ 5² = 5⁴

For m ≥ n, aᵐ ÷ aⁿ = aᵐ⁻ⁿ. The bases must match.

3

A power raised to a power

(2³)⁴ means four groups of 2³, giving twelve factors of 2:

(2³)⁴ = 2³ˣ⁴ = 2¹²

4

A product raised to a power

Expand both sides to verify the numerical relationship:

(3 × 5)² = (3 × 5)(3 × 5) = (3 × 3)(5 × 5) = 3² × 5²

Therefore (ab)² = a²b² for numerical bases.

Worked examples

Worked example 1: Simplify 7³ × 7⁵

Same base and multiplication: add indices. 7³⁺⁵ = 7⁸.

Worked example 2: Simplify 10⁹ ÷ 10⁴

Same base and division: subtract indices. 10⁹⁻⁴ = 10⁵.

Worked example 3: Simplify (4²)³

Power of a power: multiply indices. 4²ˣ³ = 4⁶.

Worked example 4: Verify (2 × 7)² = 2² × 7²

The left side is 14² = 196. The right side is 4 × 49 = 196, so the relationship is verified.

Check your understanding

1. Simplify 2⁴ × 2³.

2. Simplify 5⁸ ÷ 5³.

3. Simplify (3²)⁵.

4. Which equality correctly applies a power to a product?

5. Which expanded form proves 4² × 4³ = 4⁵?

Reason and apply

1. Simplify 6⁴ × 6⁷.

Model response: 6¹¹.

2. Simplify 9¹⁰ ÷ 9⁶.

Model response: 9⁴.

3. Simplify (5³)⁴ and explain the index.

Model response: 5¹². Four groups of three factors give 3 × 4 = 12 factors of 5.

4. A student writes 3² × 4² = 12⁴. Diagnose the error and give the correct value.

Model response: The add-indices law cannot be used because the bases differ. 3² × 4² = 9 × 16 = 144 (also (3 × 4)² = 12²).

5. Verify (4 × 6)² = 4² × 6² by evaluating both sides, then explain why it works.

Model response: The left side is 24² = 576. The right side is 16 × 36 = 576. Expanding (4 × 6)(4 × 6) and regrouping gives (4 × 4)(6 × 6).

6. Use expanded form to establish the quotient law for 3⁵ ÷ 3² and the power-of-a-power law for (2²)³.

Model response: For 3⁵ ÷ 3², cancel two of the five factors to leave 3³, so division subtracts indices. For (2²)³, three groups of two factors give six factors, so (2²)³ = 2⁶ and a power of a power multiplies indices.