Question 1: Calculate the value of 2^{-3}.

Answer: 1/8

Question 2: Evaluate 5^{-2}.

Answer: 1/25

Question 3: Simplify the expression: (3^{-2}) imes (3^{4}).

Answer: 3^{2} = 9

Question 4: Calculate the value of (1/2)^{-4}.

Answer: 16

Question 1: A bridge support uses a material that weakens over time, modeled by (0.8)^{-n}. If the weakening factor after 3 years is (0.8)^{-3}, explain what this value represents and calculate its approximate value.

Answer: Represents the weakening after 3 years; value ≈ 1.95

Question 2: An engineer uses a formula involving negative indices: V = k imes 10^{-x}. If V = 0.001 and k=1, find the value of x.

Answer: 3

Question 1: An engineer models the resistance of a material as R = 5 imes 10^{-6} imes t^{-2}, where t is time in seconds. Calculate resistance R when t=2 seconds.

Answer: R = 5 imes 10^{-6} imes (2)^{-2} = 5 imes 10^{-6} imes 1/4 = 1.25 imes 10^{-6}

Question 2: A certain component's strength decreases according to the formula S = 100 imes 10^{-0.5x}, where x is the number of years. How much strength remains after 4 years?

Answer: S = 100 imes 10^{-0.5 imes 4} = 100 imes 10^{-2} = 1

Question 1: A beam's deflection is modeled by D = k imes 10^{-x}, where for certain conditions D=0.01 when x=2. Find the constant k.

Answer: k=1

Question 2: If a pipe's flow rate Q is proportional to 10^{-m}, and Q halves when m increases by 1, find the proportionality constant and the flow rate when m=3, given Q=8 when m=0.

Answer: Q=8 imes 10^{-m}; when m=3, Q=8 imes 10^{-3} = 0.008

Question 1: Express 1/(2^{4}) in terms of negative indices.

Answer: 2^{-4}

Question 2: Evaluate (10^{-2}) imes (10^{3}).

Answer: 10^{1} = 10

Question 1: Student claims: 3^{-2} = 6. Identify and correct the error.

Answer: Incorrect; 3^{-2} = 1/9

Question 2: Student says: (2^{-3}) = -8. Explain the mistake and give the correct answer.

Answer: Incorrect; 2^{-3} = 1/8