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When you come across an algebraic expression, it often includes variables like x or y. To evaluate this expression for a specific value, substitution is your first move. This means replacing the variable with its given numerical value. Imagine you're baking a cake and the recipe asks for 'sugar'—this is the variable. If the recipe then states 'sugar = 1 cup', you'll substitute 'sugar' with '1 cup' in the instruction. Similarly, in our expression \(\frac{x^{2}}{3}\), when we learn that x equals 6, we swap the x for 6, resulting in a new expression \(\frac{6^{2}}{3}\).

After the substitution step, we carry out arithmetic operations as necessary. This might involve addition, subtraction, multiplication, or division. These are the basic building blocks of mathematics and are omnipresent in algebraic manipulations. In our example, we needed to handle exponentiation first—\(6^{2}\)—which is actually a form of repeated multiplication. Once we establish that \(6^{2} = 36\), we then move to the division part: \(\frac{36}{3}\). It's like when you split a pizza equally among friends—the size of the pizza (the numerator) divided by the number of friends (the denominator) gives you the size of each slice. Hence, \(\frac{36}{3} = 12\), which means everyone gets a slice equivalent to the number 12.

Performing these operations in the correct order is crucial. Remember the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)—to ensure accuracy in calculations.

To simplify an expression is to make it as straightforward as possible. You wouldn't give someone a complex map when a simple one would do; the same principle applies to algebra. After the arithmetic operations, we often need to reduce the expression to its simplest form. This might involve combining like terms, reducing fractions, or applying algebraic rules.

In our case, after substituting and calculating \(6^{2}\) and performing the division, which gave us 12, we've reached the simplest form of the expression. No further simplification is needed as 12 is the final answer. It is already a whole number and cannot be simplified further. In other cases, you might end up with a fraction or an algebraic term that can be further reduced, but always aim for the least complex expression that is mathematically equivalent to the original.

Remember, simplifying an expression doesn't change its value; it just makes it cleaner and more understandable, like rewriting a sentence to make it clear without changing its meaning. It's the final polish on your mathematical work.