Final answer:
The statement is false because the product of x^-2 + y^3 and x^4 - y^-2 does not necessarily result in all whole number exponents. Simply multiplying exponentiated terms does not transform negative exponents into whole numbers.
Step-by-step explanation:
The statement that the product of x^-2 + y^3 and x^4 - y^-2 will demonstrate closure because the exponents of the product are whole numbers is false. The principle of closure in mathematics dictates that an operation performed on any two elements of a set should result in an element that is also within that set. However, the concept of closure does not guarantee that the operation between elements with exponentiation will always yield exponents that are whole numbers.
When we multiply terms with exponents, we need to apply the product rule, which involves adding exponents when the bases are the same. However, since we are working with a binomial expression (two terms), we need to multiply each term in the first binomial by each term in the second binomial. In doing so, we could end up with terms that still have negative exponents. For example, multiplying x^-2 by x^4 gives us x^(4-2), which equals x^2, a whole number exponent. But multiplying x^-2 by y^-2 will yield x^-2y^-2, or 1/(x^2y^2), which does not have a whole number exponent.
Therefore, the initial assumption is incorrect because simply multiplying exponentiated terms does not ensure all resulting exponents will be whole numbers. This is due to the fact that we might still have negative exponents in some cases, and these do not automatically become positive or whole numbers through multiplication.