Question 1: Calculate the value of (3²)³.

Answer: 216

Question 2: Simplify: (x⁴)².

Answer: x⁸

Question 3: Evaluate: (2³)⁴.

Answer: 4096

Question 4: Express as a single power: (x⁵)³.

Answer: x¹⁵

Question 1: Simplify the expression: (a³)⁴ × a².

Answer: a¹⁴

Question 2: If (x²)³ = x¹⁸, what is the value of the exponent when simplifying (x²)³?

Answer: 6

Question 3: Explain why (a⁴)² ≠ a⁸ in the context of powers of powers.

Answer: Because (a⁴)² simplifies to a⁸, which is equal; the statement is correct. If it said (a⁴)³, then it would be a¹², not a⁸.

Question 4: Simplify: (xˣ)³.

Answer: x³ˣ

Question 5: Given that (b⁵)² = b¹⁰, find (b⁵)⁴.

Answer: b²⁰

Question 1: A bacteria population doubles every hour. If initially there are P bacteria, write an expression for the population after n hours using powers of powers.

Answer: P × (2ⁿ)

Question 2: A researcher models radioactive decay with the expression A = A₀ × (½)ⁿ, where n is time in hours. Rewrite this using powers of powers and simplified form.

Answer: A = A₀ × (½)ⁿ

Question 1: Simplify: (a⁻³)³.

Answer: a⁻⁹

Question 2: If (xᵐ)ⁿ = xᵐⁿ, prove that (xᵐ)ⁿ = xᵐⁿ using properties of exponents.

Answer: Using the law (aᵐ)ⁿ = aᵐⁿ, initial assumptions, and properties of exponents, the proof follows directly from the definition of powers of powers.

Question 1: Simplify: (x⁷)² × (x³)².

Answer: x¹⁴

Question 2: Express as a single power: (x²)³ × (x⁴)².

Answer: x¹²

Question 3: Identify the mistake: Simplify (a³)² = a⁶. Is this correct? Explain.

Answer: Yes, it is correct because (a³)² simplifies to a³×² = a⁶, following the law (aᵐ)ⁿ = aᵐⁿ.