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A monomial is an algebraic expression that represents a single term consisting of a constant, a variable, or a product of constants and variables with whole number exponents. It is one of the building blocks in algebra and its simplicity allows for straightforward operations such as finding the greatest common factor (GCF).

For example, in the expressions given in our exercise, \(9 y^{5}\), \(18 y^{2}\), and \(\-3 y\), each of these is a monomial. Monomials can be multiplied and divided. To find the GCF of monomials, we need to break down each term into its constituent prime factors and identify the common factors shared by each term.

Prime factorization is the process of breaking down a composite number into its prime factors, which are prime numbers that multiply together to give the original number. Prime numbers are the basic building blocks of all numbers, as they can only be divided by 1 and themselves.

In the context of the exercise, prime factorization helps us find the GCF of the given monomials. By breaking down numbers like 9 and 18 into \(3^2\) and \(2 \times 3^2\) respectively, we decompose the numbers into their simplest prime components. This step is crucial as it allows us to compare the monomials more easily and spot the common prime factors that will constitute the greatest common factor.

Exponents, also known as powers, represent how many times a number, known as the base, is multiplied by itself. It is a concise way to express repeated multiplication. In the expressions \(9 y^{5}\), \(18 y^{2}\), and \(\-3 y\), the exponent tells us how many times the variable \(y\) is used as a factor.

Understanding exponents is key to simplifying expressions and especially to calculate the GCF of monomials. In the given exercise, the exponent of \(y\) in each monomial indicates the number of times \(y\) occurs as a factor, and the lowest exponent of \(y\) in the list indicates its power in the greatest common factor. In simpler terms, if we are looking at expressions with the same base, the GCF will include that base raised to the lowest exponent present in the terms we are examining.