Question 1: Calculate the value of a⁴ × a³.

Answer: a⁷

Question 2: Simplify: 5² × 5⁶.

Answer: 5⁸

Question 3: Express as a single power: x³ × x² × x⁴.

Answer: x⁹

Question 4: Calculate: (2a⁵) × (3a³).

Answer: 6a⁸

Question 1: If aᵐ × aⁿ = aᵐ⁺ⁿ, explain why a⁵ × a² = a⁷ using the law of indices.

Answer: Because according to the law aᵐ × aⁿ = aᵐ⁺ⁿ, so a⁵ × a² = a⁵⁺² = a⁷.

Question 2: Solve for x: aˣ × a³ = a⁸.

Answer: x=5

Question 3: Given that a² × a² × a³ = aˣ, find the value of x.

Answer: x=7

Question 1: A scientist is growing bacteria. The bacteria count doubles every hour, and after 3 hours, there are a total of 8,000 bacteria. Write an expression for the bacteria count after t hours and find how many bacteria there are after 4 hours.

Answer: Number of bacteria = 1000 × 2ᵗ; after 4 hours: 1000 × 2⁴ = 16,000

Question 2: A piece of radioactive material decays such that its remaining mass after t days is modeled by m = m₀ × (1/2)ᵗ. If the initial mass m₀ is 16 grams, how much remains after 5 days?

Answer: Mass after 5 days = 16 × (1/2)⁵ = 16/32 = 0.5 grams

Question 1: Prove that (a²b³) × (a³b²) = a⁵b⁵ using the law of indices.

Answer: a²b³ × a³b² = a²⁺³ × b³⁺² = a⁵b⁵

Question 2: If a⁻³ × a⁵ = aˣ, find x and explain the steps.

Answer: x=2 because a⁻³ × a⁵ = a⁻³⁺⁵ = a²

Question 1: Incorrect statement: a⁴ × a⁻² = a². Find the mistake and provide the correct answer.

Answer: Mistake: a⁴ × a⁻² = a⁴⁻² = a² (which is correct), so the statement is actually correct; trick question.

Question 2: A student writes: a⁶ ÷ a² = a⁸. Is this correct? If not, correct it.

Answer: Incorrect, because a⁶ ÷ a² = a⁶⁻² = a⁴.