Question 1: Evaluate the expression 3^0.

Answer: 1

Question 2: Simplify the expression (2^0) × (5^0).

Answer: 1

Question 3: State and prove the rule for any non-zero base a: a^0 = ?

Answer: a^0 = 1

Question 4: Calculate the value of 7^0 + 0^0. (Note: 0^0 is an indeterminate form, but in many contexts, it's considered 1 for this exercise.)

Answer: 2

Question 5: Explain why any non-zero number raised to the zero power equals 1, using a proof based on the laws of exponents.

Answer: Using the law a^m / a^n = a^{m-n}, setting m=n gives a^n / a^n = 1, so a^0 = 1.

Question 6: Plot the graph of y = x^0 for x = -3 to 3 on the grid.

Answer: A horizontal line at y=1 across the x-range.

Question 7: Construct a triangle on the grid with sides of length 5 units, using the zero index rule to determine the length of a side when raised to the zero power.

Answer: All sides are 5 units; the zero index rule relates to exponents, not sides directly.

Question 8: A student claims that 0^0 = 0. Is this correct? Explain your reasoning.

Answer: No, in most mathematical contexts, 0^0 is defined as 1, although it is an indeterminate form in calculus.

Question 9: If a ≠ 0, prove that a^{m} / a^{n} = a^{m-n} using the properties of exponents, with a focus on the case where m=n=0.

Answer: a^{m} / a^{n} = a^{m-n} by the law of exponents. When m=n=0, a^0 / a^0 = 1/1=1, confirming a^0=1.

Question 10: Identify the common mistake in the following statement: 'Any number raised to the zero power equals zero.'

Answer: The mistake is that any non-zero number raised to zero power equals 1, not zero.

Question 11: Challenge: Find all solutions to the equation a^0 = 1 for any real number a.

Answer: All real numbers a ≠ 0 satisfy a^0 = 1.