91Ó°ÊÓ

At their core, algebraic expressions are combinations of numbers, variables (like x or y), and arithmetic operations such as addition, subtraction, multiplication, and division. In the given exercise, \(x^{2} y^{2}-5\)^{2} is an example of an algebraic expression that involves a variable raised to a power, in this case squared, and a constant.

To work with such expressions, it's crucial to understand the operations involved and the order in which they should be executed, often guided by the rules of exponents and the distributive property. Specifically, in the expression \(x^{2} y^{2}-5\)^{2}, we are looking at the square of a binomial, which can be expanded using the binomial theorem or specific binomial formulas.

The process of simplifying expressions reduces them to their most elementary form, making them easier to understand or further manipulate. When we simplify, we combine like terms, apply the order of operations, and use algebraic rules to rewrite expressions in a simpler way.

In the context of our problem, simplifying the binomial expansion involves squaring both a and b, and doubling the product of a and b. The initial expansion based on the formula \(a-b)^{2} = a^{2} - 2ab + b^{2}\) gives us a raw expanded form. To simplify, we perform the required exponentiation and multiplication, ultimately combining all the like terms possible to achieve the final, simplest form of the expression: \(x^{4} y^{4} - 10x^{2} y^{2} + 25\).

When we talk about polynomial multiplication, we refer to the process of multiplying two or more polynomials together. A polynomial is an algebraic expression that can have constants, variables, and exponents, which are combined using arithmetic operations.

Polynomial multiplication can sometimes involve the application of the distributive property in a more extensive manner, especially when dealing with binomials or larger polynomials. The distributive property is also at play when we use specific formulas for binomial expansion, such as the one in our exercise. Knowing the relevant formulas and how to apply them can greatly simplify the process of multiplying polynomials.

The step-by-step solution shows how we multiply the polynomial \(a-b\) by itself. We square each term (\textbf{note}: squaring a term is equivalent to multiplying the term by itself) and then apply the middle term formula (which in this case involves subtraction), to achieve the expanded polynomial.