Question 1: Simplify the expression: (x³)⁴

Answer: x¹²

Question 2: Calculate the value of: (2²)³

Answer: 8

Question 3: Simplify: (5⁴)²

Answer: 5⁸

Question 4: Express (x⁵)⁰ in simplified form.

Answer: 1

Question 1: Explain whether the expression (a²)³ is equal to a⁶. Show your reasoning.

Answer: Yes, because (a²)³ = a² * 3 = a⁶ according to the law (aᵐ)ⁿ = aᵐⁿ.

Question 2: A cube has side length 3x. Write an expression for the volume of the cube using indices, and simplify using the law (aᵐ)ⁿ = aᵐⁿ.

Answer: Volume = (3x)³ = 3³ * x³ = 27x³

Question 3: If (a⁴)² = aᵧ, what is the value of y?

Answer: 8

Question 4: Identify and correct the mistake in simplifying (x⁵)² as x¹².

Answer: The mistake is applying the wrong exponent rule. (x⁵)² = x⁵ * 2 = x¹⁰, not x¹².

Question 1: A bacteria population doubles every hour. If the initial population is P, write an expression for the population after n hours, assuming the growth follows the law (aᵐ)ⁿ = aᵐⁿ. Simplify your expression.

Answer: Population = P * 2ⁿ

Question 1: Prove that (a²)³ = a⁶ using the laws of indices, and discuss common misconceptions that might lead to errors.

Answer: Using (aᵐ)ⁿ = aᵐⁿ, (a²)³ = a² * 3 = a⁶. Common misconception: confusing multiplication of exponents with addition, e.g., wrongly concluding (a²)³ = a² + 3.

Question 2: Given that (x⁴)² = x⁸, find the value of n in the expression (x⁴)ⁿ that equals x¹².

Answer: 3

Question 1: Simplify: (x⁶)⁰

Answer: 1

Question 2: Express the following as a single power: (3²)³

Answer: 3⁶

Question 3: If (a³)² = a⁸, is this correct? Explain briefly.

Answer: No, because (a³)² = a⁶, not a⁸. The correct rule gives aᵐⁿ = aᵐⁿ, so (a³)² = a⁶.