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

Answer: 1/8

Question 2: Evaluate the expression: 5^{-2} + 3^{-1}.

Answer: 1/25 + 1/3

Question 3: Simplify: (4^{-1})^2.

Answer: 1/16

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

Answer: 4

Question 1: Evaluate 3^{-2} + 2^{-3} and explain each step.

Answer: 1/9 + 1/8

Question 2: A calculator shows 10^{-3}. How does this value relate to the number 0.001? Explain why.

Answer: Because 10^{-3} = 1/10^{3} = 1/1000 = 0.001

Question 3: Construct an expression involving negative indices that equals 1/16. Show your reasoning.

Answer: 2^{-4}

Question 4: Explain why (2^{-3})^{-2} equals 2^{6}. Show your working.

Answer: Using the power of a power rule: (a^{m})^{n} = a^{m*n}. So, (2^{-3})^{-2} = 2^{-3 * -2} = 2^{6}.

Question 1: A research lab uses a microscope with a zoom factor of 10^{-4} to view tiny particles. What does this negative index tell you about the zoom level? Explain your reasoning.

Answer: It indicates the zoom level is 1/10,000, meaning the microscope magnifies objects up to 10,000 times smaller than their actual size.

Question 2: An engineer calculates that a certain component has a resistance of 10^{-9} ohms. What does the negative exponent imply about this resistance? Explain.

Answer: It implies the resistance is in the nanohm range, very small, as 10^{-9} equals 1 nanohm.

Question 1: Simplify: (8^{-2} imes 4^{3}) / 2^{-4}. Show every step.

Answer: 1/64 * 64 / 16 = 1/16

Question 2: Prove that 2^{-n} = 1 / 2^{n} for any positive integer n, using evaluation and explanation.

Answer: By definition, negative exponents indicate reciprocal: 2^{-n} = 1 / 2^{n}.

Question 1: Student's answer: 3^{-2} = 3^{2} = 9. Find and explain the mistake.

Answer: The mistake is assuming 3^{-2} = 3^{2}. Correct answer is 1/9 because negative exponent indicates reciprocal.

Question 2: Student's calculation: (1/5)^{-3} = 5^{3} = 125. Identify the mistake and correct it.

Answer: The mistake is ignoring the negative exponent. Correct is (1/5)^{-3} = 5^{3} = 125, which is correct. But explanation should clarify that (1/5)^{-3} = (5^{-1})^{-3} = 5^{3}.