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When we talk about simplifying algebraic expressions, we're referring to the process of making an expression easier to understand or work with. This is often achieved by combining like terms and reducing expressions to their simplest form. In the given exercise, simplifying involves multiplying the coefficients and the variables separately and then combining them to form a single expression.

For example, when given \(5 m^{2} n \times 3 m n^{2}\), we look for the terms that are alike, which are the variable terms with the same bases. The coefficients (5 and 3) are multiplied together to give us a single numerical value, and the variables are simplified by using exponent rules, which we will discuss in detail in the next section. The goal of simplification is to ensure that there is no further multiplication or addition that can be done within the expression.

Understanding exponent rules is crucial when working with polynomials, especially when simplifying expressions that involve powers. There are a few basic rules that you should remember:

• When multiplying terms with the same base, you add the exponents (\(a^m \times a^n = a^{m+n}\)).
• When raising a power to a power, you multiply the exponents (\( (a^m)^n = a^{m \times n}\)).
• When dividing terms with the same base, you subtract the exponents (\(a^m \text{/} a^n = a^{m-n}\)).

In the worked-out exercise, we are applying the first rule. We have \(m^2\) and \(m\) which makes \(m^{2+1}\) when combined, and \(n\) and \(n^2\) which makes \(n^{1+2}\) following the same rule. These exponent rules significantly simplify the process of multiplying polynomials.

The distributive property is another essential concept used when multiplying expressions, especially polynomials. It allows us to multiply a single term by each term within a parenthesis. In algebraic form, the distributive property is represented as \(a(b + c) = ab + ac\). Although in this problem, we don't have addition or subtraction, we still use the distributive property to multiply two binomials.

In the given exercise, we use this property to distribute the multiplication across the terms in each polynomial. First, we multiply the coefficients and then distribute the multiplication of the variables. Following the distributive property ensures that no term is left out, and the multiplication is conducted orderly and correctly, leading to a simplified and accurate solution.