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Understanding how to simplify expressions is crucial in mathematics. It involves reducing a complex expression into its simplest form using various mathematical operations. The process ensures that expressions are more manageable and easier to work with, especially in solving equations and evaluating numerical values.

When simplifying, it's essential to follow a set order of operations to avoid mistakes. Mistakes in the order can lead to incorrect results, even if the mathematical operations are performed correctly. Simplification can involve combining like terms, using the distributive property, and canceling common factors when dealing with fractions.

The given problem involves a fractional numerical expression where simplification is achieved by performing multiplication and addition on the numerator before dividing by the denominator. It's a straightforward example of how a proper order of operations can lead to the correct simplified result.

A numerical expression is a mathematical statement involving only numbers and operation symbols. It does not contain any variables and is solely comprised of digits and arithmetic operations such as addition, subtraction, multiplication, division, and occasionally exponents.

To clarify, a numerical expression represents a single numeric value once it is fully simplified. For example, in the exercise provided, \( \frac{8+5 \cdot 2}{12} \) is a numerical expression. We approach this expression with the aim of simplifying it to find that single numeric value it represents. This process of simplification is critical as it lays the foundation for more complex problem-solving techniques in algebra and beyond.

PEMDAS is an acronym that helps remember the order of operations in mathematics. It stands for Parentheses, Exponents (including roots, such as square roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule is fundamental in ensuring that every mathematician speaks the same 'mathematical language' when it comes to evaluating expressions.

When simplifying expressions, always start by solving operations inside parentheses first, then move on to exponents. Next, perform any multiplication and division as they appear from left to right in the expression. Finally, conclude with addition and subtraction in the order they appear from left to right. In our exercise, we follow these steps: there are no parentheses or exponents, so we start with multiplication, then add the results to the numerator, and finally, divide by the denominator to get the simplified expression.