Question 1: Express 81 as a power of 3 using index notation.

Answer: 3^4

Question 2: Simplify the expression: 2^5 × 2^3.

Answer: 2^8

Question 3: Write 120 in its prime factorization form using indices.

Answer: 2^3 × 3 × 5

Question 4: If a^3 = 125, find the value of a in index notation.

Answer: 5

Question 5: Explain why 2^4 × 2^(-2) = 2^(4 + (-2)).

Answer: Using the law of indices that a^m × a^n = a^{m + n}

Question 6: Calculate and simplify: (3^2)^4.

Answer: 3^8

Question 7: Construct a triangle on the grid with sides of length 2^3 and 2^4, meeting at a point. Indicate the lengths in index notation.

Answer: 3 and 4 (or 2^3 and 2^4)

Question 8: Express the product 16 × 81 using index notation with prime bases.

Answer: 2^4 × 3^4

Question 9: Identify and correct the mistake: Simplify 2^5 / 2^3 = 2^2.

Answer: The correct answer should be 2^{5-3} = 2^2, so the mistake is in the expression or calculation if any, but the given is correct; students might explain that division of indices is subtraction.

Question 10: Using the laws of indices, simplify: (2^3)^2 × 2^4.

Answer: 2^{6 + 4} = 2^{10}

Question 11: A calculator shows 2^10 = 1024. Write 1024 as a power of 2 in index notation.

Answer: 2^{10}