Question 1: Calculate a⁵ ÷ a² and express your answer in simplified form.

Answer: a³

Question 2: Simplify the expression a⁸ ÷ a⁴.

Answer: a^4

Question 3: Express the result of a¹⁰ ÷ a⁷ in exponential form.

Answer: a³

Question 4: Calculate a⁶ ÷ a³ and write the answer in simplest form.

Answer: a³

Question 1: Incorrect: a⁴ ÷ a² = a² + a² = a⁴. What is wrong with this reasoning?

Answer: The mistake is adding the exponents; according to index laws, division subtracts exponents, not adds.

Question 2: Incorrect: a³ ÷ a⁵ = a³ - a⁵ = a⁻². Is this correct? Explain.

Answer: No, the operation is correct, but students sometimes forget that the result is a⁻², which indicates a reciprocal.

Question 3: Identify the mistake: Simplifying a⁷ ÷ a³ as a⁷ + a³.

Answer: The mistake is adding exponents; correct way is subtracting exponents: a⁷ ÷ a³ = a⁷⁻³ = a⁴.

Question 1: If a⁸ ÷ a⁴ = a⁴, what is the value of a if a ≠ 0? Explain your reasoning.

Answer: a can be any non-zero number because the law holds for all such a; the division simplifies to a⁸⁻⁴ = a⁴.

Question 2: Simplify the expression a² ÷ a⁻³ and interpret the result.

Answer: a² ÷ a⁻³ = a²⁻⁻³ = a²⁺³ = a⁵

Question 1: A bacteria population doubles every hour. If initially there are a bacteria, express the population after t hours as a power of a, assuming the initial population is a. How does dividing the populations over different times relate to index laws?

Answer: The population after t hours is a×2^t. Dividing populations corresponds to subtracting exponents of 2, similar to index law: 2^m ÷ 2^n = 2^{m−n}.

Question 1: Prove that for any non-zero a, a⁰ = 1 using the law aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

Answer: a⁰ = a⁰⁻⁰ = a⁰, and since any number to the zero power equals 1, a⁰ = 1.

Question 2: If a⁵ ÷ a³ = a², find the value of (a²)⁴ and express it in terms of a.

Answer: a^{2×4} = a^8

Question 1: Simplify: a³ ÷ a⁵.

Answer: a⁻²

Question 2: Express as a single power: (a²)(a³) ÷ a⁴.

Answer: a^{2+3−4} = a¹