Index Laws with Positive Indices Worksheet

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?

6. Simplify 6⁴ × 6⁷.

7. Simplify 9¹⁰ ÷ 9⁶.

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

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

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

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

Answers and reasoning

1. Answer

2⁷. Same-base multiplication adds the indices: 4 + 3 = 7.

2. Answer

5⁵. Same-base division subtracts the indices: 8 − 3 = 5.

3. Answer

3¹⁰. A power of a power multiplies the indices: 2 × 5 = 10.

4. Answer

(2 × 5)² = 2² × 5². Squaring the product creates two factors of both 2 and 5, so (2 × 5)² = 2² × 5².

5. Answer

(4 × 4)(4 × 4 × 4). Together there are five factors of 4.

6. Model response

6¹¹.

7. Model response

9⁴.

8. Model response

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

9. Model response

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

10. 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).

11. 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.