Final answer:
Joseph is correct in stating that the question mark should be replaced by -5. This is a consequence of the properties of exponents where any exponent raised to the zero power equals 1, and when multiplying like bases, their exponents are added. Therefore, x^5 multiplied by x^-5 equals x^0, which is 1.
Step-by-step explanation:
Mr. Morris's class is tasked with solving the following problem: (x^3)^0 = x^5 * x^? Among the students, Joseph's answer is correct. The properties of exponents inform us that any number raised to the zero power, such as (x^3)^0, is equal to 1. Additionally, when we multiply two exponential expressions with the same base, we add the exponents according to Eq. A.8. Here we have x^5 * x^?, and we need the result to be 1 since (x^3)^0 is 1. Therefore, the exponent question mark needs to be such that adding it to 5 results in 0, which is where the -5 comes in. We have x^5 * x^-5 = x^(5 + -5) = x^0, which is 1. Thus, the correct solution is for the question mark to be replaced by -5.
Using the property that tells us x^-n = 1/x^n, where negative exponents denote a division, we can see that x^-5 is equivalent to 1/x^5. Multiplying this by x^5 yields a product of 1, confirming Joseph's answer.