Question 1: Calculate 2^4.

Answer: 16

Question 2: Evaluate 3^3.

Answer: 27

Question 3: Simplify: 2^3 × 2^4.

Answer: 2^7

Question 4: What is the value of 3^0?

Answer: 1

Question 1: Student's mistake: 2^3 = 8, so 2^3 + 2^3 = 16. Is this correct? Correct the error.

Answer: No, 2^3 + 2^3 = 2^4 (not 16). The correct calculation is 2^3 + 2^3 = 2 × 2^3 = 2^4.

Question 2: A student says: 3^2 = 9, so 3^2 × 3^3 = 3^5. Is this correct? Explain and correct if needed.

Answer: Yes, because when multiplying powers with the same base, add exponents: 3^2 × 3^3 = 3^(2+3) = 3^5.

Question 1: Calculate the value of 4^3 + 2^5 and explain your steps.

Answer: 4^3 = 64, 2^5 = 32, so total = 64 + 32 = 96.

Question 2: If 5^x = 125, find x and justify your answer.

Answer: Since 125 = 5^3, x = 3.

Question 1: A bacteria population doubles every hour. If it starts with 1 bacteria, how many bacteria are there after 4 hours? Express your answer using powers of 2.

Answer: After 4 hours, the population is 2^4 = 16 bacteria.

Question 1: Prove that 2^a × 2^b = 2^{a+b} for integers a and b. Write a short explanation and example.

Answer: Using index laws: 2^a × 2^b = 2^{a + b}. Example: 2^2 × 2^3 = 2^{2+3} = 2^5 = 32.

Question 2: Express 128 in terms of powers of 2. Show your working.

Answer: 128 = 2^7, since 2^7 = 128.

Question 1: Calculate 3^4 ÷ 3^2.

Answer: 3^2

Question 2: What is 2^5 + 2^5?

Answer: 2 × 2^5 = 2^6

Question 3: Express the number 16 as a power of 2.

Answer: 16 = 2^4