Negative Index Laws Proofs

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📚 Part 1: Daily Review - Positive Index Laws

1. Which index law is represented by a³ × a⁵ = a⁸?

Addition law: a^m × aⁿ = a^m+n

Subtraction law: a^m ÷ aⁿ = a^m-n

Power law: (a^m)ⁿ = a^mn

2. Simplify: 2⁴ ÷ 2⁴ = ____________

Hint: What happens when you divide identical terms?

3. Using index laws, what is a⁵ ÷ a⁷?

a²

a^-2

a¹²

🔍 Part 2: Developing Negative Index Proofs

4. WALT: Develop and test mathematical proofs for index laws involving negative indices.

Success Criteria: I can explain negative indices using mathematical reasoning and construct step-by-step proofs.

5. Complete this proof that a^-m = 1/a^m:

Step 1: Start with a^m ÷ a^m = ____________

Step 2: Using index laws: a^m ÷ a^m = a^m-m = a^___

Step 3: We know a⁰ = ____________

Step 4: Now consider a⁰ ÷ a^m = a^0-m = a^___

Step 5: But a⁰ ÷ a^m = 1 ÷ a^m = ____________

Step 6: Therefore: a^-m = ____________

6. Use your proof to explain why 3^-2 = 1/9:

7. Extension Challenge: Prove that (a^-2)³ = a^-6 using the power law:

✏️ Part 3: Application and Reflection

8. Simplify these expressions using negative index laws:

a) 5^-3 = ____________

b) x⁴ × x^-7 = ____________

c) (2y)^-2 = ____________

9. Check all the statements that are true:

4^-1 = -4

4^-1 = 1/4

x^-3 × x³ = 1

(ab)^-2 = 1/(a²b²)

10. Exit Ticket: Explain why 2^-3 = 1/2³ using your own words or a brief proof outline:

Negative Index Laws Proofs

Negative Index Laws Proofs

📚 Part 1: Daily Review - Positive Index Laws

🔍 Part 2: Developing Negative Index Proofs

✏️ Part 3: Application and Reflection

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