Question 1: Calculate the value of 3³ × 3⁴ using the law of indices.

Answer: 3⁷

Question 2: Simplify the expression: 5² × 5³.

Answer: 5⁵

Question 3: If aᵐ × aⁿ = aᵐ⁺ⁿ, what is 2⁴ × 2⁶?

Answer: 2¹⁰

Question 1: Express 7³ × 7² as a single power of 7.

Answer: 7⁵

Question 2: Simplify the expression: 4⁵ × 4³.

Answer: 4⁸

Question 3: Calculate the value of 6² × 6⁴.

Answer: 6⁶

Question 1: A plant grows such that its height doubles every month. If it starts at 2 meters, express its height after 3 months using indices, and explain how the law of indices applies.

Answer: 2 × 2³ = 2¹ × 2³ = 2⁴; the height after 3 months is 2⁴ meters because multiplying powers with the same base adds exponents.

Question 2: A scientist increases the number of bacteria by a factor of 10 each hour. Initially, there are 5 bacteria. Write an expression for the number of bacteria after n hours and simplify.

Answer: 5 × 10ⁿ

Question 1: A bacteria culture starts with 1,000 bacteria and doubles every 2 hours. Write an expression for the number of bacteria after t hours, and determine the number after 6 hours.

Answer: Number of bacteria = 1000 × 2^{t/2}; after 6 hours: 1000 × 2^{6/2} = 1000 × 2³ = 8000

Question 1: If aᵐ × aⁿ = aᵐ⁺ⁿ, find the value of m and n if a³ × aⁿ = a¹⁰ and aᵐ × a² = a⁸.

Answer: m = 8, n = 7

Question 2: Construct an expression to model the total number of particles if the initial number is 3 and each step multiplies the count by a^{2}. What is the total after 4 steps?

Answer: Initial: 3; after 4 steps: 3 × (a²)^4 = 3 × a⁸

Question 1: Identify and correct the mistake in simplifying: 2⁴ × 2⁵ = 2⁹ - 1.

Answer: Mistake: subtraction sign; correct: 2⁹

Question 2: Explain why the expression 5⁶ × 5² = 5⁸ is correct or incorrect.

Answer: Correct; since adding exponents: 5⁶ × 5² = 5^{6+2} = 5⁸.