Question 1: Calculate the value of 3^4.

Answer: 81

Question 2: Simplify the expression: 2^3 × 2^4.

Answer: 2^7

Question 3: Evaluate: (5^2)^3.

Answer: 15625

Question 4: Express 64 as a power of 2.

Answer: 2^6

Question 1: If a computer's memory is 2^30 bytes, how many gigabytes (GB) is this, given that 1 GB = 2^30 bytes? Show your reasoning.

Answer: 1 GB

Question 2: A bacteria population doubles every 3 hours. If initially there are 2^4 bacteria, how many bacteria will there be after 12 hours? Write your evaluation process.

Answer: 2^4 × 2^(12/3) = 2^4 × 2^4 = 2^8 = 256 bacteria

Question 3: Evaluate the expression: 4^3 ÷ 2^4.

Answer: 4^3 ÷ 2^4 = (2^2)^3 ÷ 2^4 = 2^6 ÷ 2^4 = 2^{6-4} = 2^2 = 4

Question 1: A satellite's signal strength decreases proportional to the square of the distance from Earth. If at 1000 km the signal strength is 1 unit, what is the signal strength at 2000 km? Show your evaluation.

Answer: At 2000 km, signal strength = 1 / (2^2) = 1/4 units

Question 2: A digital file size is 2^20 bytes. How many kilobytes (kB) is this, given 1 kB = 2^10 bytes? Show your work.

Answer: 2^20 bytes ÷ 2^10 = 2^{20-10} = 2^{10} = 1024 kB

Question 1: Express the result of (3^4 × 3^2) ÷ 3^5 as a single power of 3.

Answer: 3^(4+2-5) = 3^1 = 3

Question 2: A robot's battery capacity is rated at 2^12 mAh. If the robot uses 2^8 mAh per hour, how many hours can it operate before the battery is drained? Show your calculation.

Answer: 2^12 mAh / 2^8 mAh per hour = 2^{12-8} = 2^4 = 16 hours

Question 1: Evaluate: (2^3)^4.

Answer: 2^{3×4} = 2^{12}

Question 2: Simplify: 10^5 ÷ 10^2.

Answer: 10^{5-2} = 10^3

Question 1: A student evaluates 2^3 + 2^3 as 2^{3+3} = 2^6. Explain the mistake and provide the correct evaluation.

Answer: The student incorrectly added powers instead of multiplying. Correct way: 2^3 + 2^3 = 2×2^3 = 2^{1+3} = 2^4 = 16