Question 1: Simplify the expression: 2^{-3}

Answer: 1/8

Question 2: Calculate the value of: (3^{-2}) imes (3^{4})

Answer: 27

Question 3: Express 5^{-1} as a fraction.

Answer: 1/5

Question 4: Simplify: (1/4)^{-2}

Answer: 16

Question 1: Evaluate the expression: 2^{-2} + 3^{-1} imes 4^{-2}. Explain each step of your calculation.

Answer: 0.25 + 1/3 imes 1/16 = 0.25 + 1/48 ≈ 0.2708

Question 2: A scientist reduces a substance's quantity by a factor of 10 each day, modeled by 10^{-n}. If after 3 days, the remaining amount is 0.001, what was the original amount? Show your reasoning.

Answer: Original amount = 10^{3} = 1000

Question 3: Express the product of 2^{-3} and 8^{-2} as a single power of 2.

Answer: 2^{-3} imes (2^{3})^{-2} = 2^{-3} imes 2^{-6} = 2^{-9}

Question 4: Construct an expression to evaluate: (1/2)^{-4} ÷ (1/4)^{-2}. Show your working.

Answer: 2^{4} ÷ 4^{2} = 16 ÷ 16 = 1

Question 1: A radioactive substance has a half-life described by 2^{-n} days. If a sample reduces to 1/16 of its original amount, find the value of n. Explain your reasoning.

Answer: Since 2^{-n} = 1/16 = 2^{-4}, n=4

Question 2: An engineer models the resistance of a material by R = 5^{-k} ohms, where k is a positive integer. If the resistance is 1 ohm, what is the value of k?

Answer: k=0

Question 1: If 3^{x} = 3^{-2} imes 3^{4}, find the value of x without using a calculator. Explain your steps.

Answer: x = -2 + 4 = 2

Question 2: Evaluate the value of (7/2)^{-3} imes 2^{3} and simplify your answer.

Answer: (2/7)^{3} imes 2^{3} = (8/343) imes 8 = 64/343

Question 1: Calculate: (2^{-2})^3. Common mistake: students forget to multiply the exponents. Identify and correct the mistake.

Answer: 2^{-6}

Question 2: Simplify: (1/3)^{-2} + (1/3)^{-3}. Explain why adding these two terms is valid or invalid, and provide the correct approach if necessary.

Answer: Both are positive powers: 3^{2} + 3^{3} = 9 + 27 = 36. Adding is valid because they are like terms after rewriting.