Question 1: Calculate 2^0.

Answer: 1

Question 2: Simplify: 5^0 × 7^0.

Answer: 1

Question 3: Evaluate: (3^0)^4.

Answer: 1

Question 4: Express 1 as a power of 10.

Answer: 10^0

Question 1: A student claims that any number raised to the zero power is zero. Is this correct? Explain your reasoning and provide a correct statement about zero indices.

Answer: No, the statement is incorrect. The rule for zero index states that for any non-zero number a, a^0 = 1. This is based on the laws of exponents, specifically a^n / a^n = a^{n-n} = a^0, which equals 1 for a ≠ 0.

Question 2: Simplify the expression: (2^3) / (2^0).

Answer: 8

Question 3: If a = 5, what is the value of a^0? Explain why.

Answer: a^0 = 1 because any non-zero number raised to the zero power equals 1, based on exponent rules.

Question 4: Prove that 7^0 = 1 using exponent laws.

Answer: Using the law a^m / a^n = a^{m-n}, set m = n, so 7^n / 7^n = 7^{n-n} = 7^0. Since 7^n / 7^n = 1, it follows that 7^0 = 1.

Question 1: A scientist records the number of bacteria as 3^0 in a petri dish. What does this number indicate? Explain your reasoning.

Answer: The number indicates 1 bacteria. Since 3^0 = 1, it shows that regardless of the base, the zero index always equals one, representing a single unit.

Question 1: Given that 4^a × 4^b = 4^{a+b}, find the value of a if 4^a = 1 and explain your reasoning.

Answer: a = 0, because 4^a = 1 implies a = 0, based on the zero index rule.

Question 2: Construct a mathematical proof to show that for any non-zero base a, a^0 = 1.

Answer: Using the exponent law a^m / a^m = a^{m-m} = a^0, since a^m / a^m = 1, it follows that a^0 must be 1, provided a ≠ 0.

Question 1: A student says: 'Any number raised to the zero power is zero.' Correct their statement and explain the misconception.

Answer: The student is incorrect. The correct statement is: Any non-zero number raised to the zero power is 1. The misconception is confusing the zero exponent with the zero value.

Question 2: Identify the mistake: 10^0 = 0.

Answer: The mistake is that 10^0 = 1, not 0. The zero index rule states any non-zero base raised to zero equals 1.