Question 1: Express 8 × 2³ using index notation.

Answer: 2^4

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

Answer: 3^6

Question 3: Rewrite 125 as a power of 5 in index notation.

Answer: 5^3

Question 1: If 2^x = 16, express x in index notation. Then, find the value of 2^x.

Answer: 4

Question 2: A number is expressed as 3^2 × 3^3. Write it as a single power of 3 and calculate its value.

Answer: 3^5, 243

Question 1: A bacteria population doubles every hour. Write an expression using index notation for the population after t hours if the initial population is 1.

Answer: 2^t

Question 1: Simplify the expression: (2^3 × 3^2)^2.

Answer: 2^6 × 3^4

Question 2: Express the number 1,296 as a product of prime factors with exponents, then write it using index notation.

Answer: 2^4 × 3^4

Question 1: Identify the mistake in the following: 5^3 × 5^2 = 5^5 (Correct).

Answer: The mistake is that 5^3 × 5^2 = 5^{3+2} = 5^5, which is correct; the statement is correct.

Question 2: Write 64 as a power of 2 and simplify: 2^6 × 2^3.

Answer: 2^9

Question 1: A student writes: 4^3 = 2^6, but then claims 4^3 = 2^5. Identify the mistake and correct it.

Answer: Incorrect: 4^3 = (2^2)^3 = 2^{2×3} = 2^6. The mistake is claiming 4^3 = 2^5. Correct answer: 4^3 = 2^6.

Question 2: A student confuses the rule: 2^a × 2^b = 2^{a+b} but writes 2^{a×b} instead. Explain the error and how to fix it.

Answer: Error: multiplying exponents is incorrect; the rule states add exponents. Correct: 2^a × 2^b = 2^{a+b}.