Multiplication and Division Index Laws — Solutions

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1. Multiplication index law

   1. 2⁷
   2. 5⁸
   3. 3⁸
   4. 7⁶
   5. 10⁸
   6. 4⁸
   7. 6⁸
   8. 9⁵

2. Division index law

   1. 5⁴
   2. 2⁵
   3. 6⁶
   4. 10⁵
   5. 3⁴
   6. 7⁰ = 1
   7. 4⁵
   8. 8³

3. Mixed multiplication and division

   1. 2⁴
   2. 3⁵
   3. 5³
   4. 10²
   5. 4²
   6. 7⁷
   7. 6³
   8. 2⁶

4. Missing index

   1. □ = 4
   2. □ = 4
   3. □ = 7
   4. □ = 7
   5. □ = 0
   6. □ = 4 (each)
   7. □ = 4
   8. □ = 3

5. True or False

   1. 3⁴ × 3² = 3⁸: False — 3⁶
   2. 5⁶ ÷ 5² = 5³: False — 5⁴
   3. 2³ × 4³ = 8⁶: False — 2³ × 4³ = 8 × 64 = 512; but 8⁶ = 262 144. Different bases — the multiplication index law does not apply.
   4. 7⁵ ÷ 7⁵ = 0: False — 7⁰ = 1
   5. 6⁴ × 6⁰ = 6⁴: True
   6. 10³ × 10² = 100⁵: False — 10⁵

6. Problem solving

   1. (2⁴ × 2³) ÷ (2⁵ ÷ 2²): 2⁴
   2. 3⁵ × 3⁴: 3⁹ = 19 683
   3. Jake’s claim 2³ × 3⁴ = 6⁷: Incorrect — different bases; answer is 648
   4. Width = 5⁹ ÷ 5⁴: 5⁵ = 3125 m

7. Three or more index terms

   1. 3² × 3³ × 3⁴: 3⁹
   2. 5⁸ ÷ 5³ ÷ 5²: 5³
   3. 2¹ × 2² × 2³ × 2⁴: 2¹⁰
   4. 7¹⁰ ÷ 7⁴ ÷ 7²: 7⁴
   5. 4³ × 4⁰ × 4⁵: 4⁸
   6. 6⁹ ÷ 6³ ÷ 6⁰: 6⁶
   7. 9² × 9³ × 9¹: 9⁶
   8. 10¹² ÷ 10⁴ ÷ 10⁵: 10³

8. Write and simplify from descriptions

   1. 4³ × 4⁶: 4⁹
   2. 7⁹ ÷ 7⁴: 7⁵
   3. 2⁵ × 2³ ÷ 2⁶: 2²
   4. 3⁸ ÷ (3² × 3³): 3³

9. Comparing simplified results

   1. 2⁵ × 2³ vs 2⁴ × 2⁴: Equal — both 2⁸
   2. 3⁹ ÷ 3³ vs 3⁷ ÷ 3¹: Equal — both 3⁶
   3. 5⁴ × 5² vs 5³ × 5³: Equal — both 5⁶
   4. 4⁷ ÷ 4³ vs 4⁶ ÷ 4²: Equal — both 4⁴

10. Real-world multi-step

    1. Bacteria on Friday: 2⁸ = 256 bacteria
    2. Warehouse trips: 3⁵ = 243 trips
    3. 6⁵ × 6³ ÷ 6⁴: Aiden is correct — 6⁴