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Understanding the order of operations is a fundamental skill in algebra. It refers to the rules that determine the sequence in which different parts of a mathematical expression are evaluated. The common acronym to remember this sequence is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

For example, in the expression \( -a^{2} \), the order of operations tells us that we need to handle the exponentiation before applying any negation. This is why, when \( a = 2 \), we first calculate \( 2^{2} \) which gives us \( 4 \), and then apply the negative sign, resulting in \( -4 \).

It's essential to follow the order of operations strictly to avoid errors in calculation. Misplacing these steps can lead to completely different results and is a common mistake for students learning algebra.

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent tells us how many times the base is to be multiplied by itself. In the context of our problem, the base is \( a \), and the exponent is \( 2 \).

When \( a=2 \), exponentiation means we need to perform \( 2 \times 2 \) which equals \( 4 \). It is critical to understand that when a negative sign is in front of the base without parentheses, as in \( -a^{2} \), the exponentiation applies only to the base, not to the negative sign. This is a source of frequent confusion, but remembering that exponents apply to the number directly next to them—and not the sign—can help clarify the process.

In our problem, exponentiation is the second step, right after substituting the value of \( a \), according to the order of operations.

Negation in algebra involves changing the sign of a number. In the expression \( -a^{2} \), the negative sign before the base indicates a negation of the number that will be the result of \( a^{2} \). This is where order of operations becomes very important; as seen in the problem, exponentiation should be performed first, and only then should negation be applied.

It's a common misconception to think that \( -a^{2} \) and \( -(a^{2}) \) are the same; however, they are not. The former negates the base after exponentiation, while the latter applies the negation to the entire value after exponentiation. When dealing with negation, it's essential to pay close attention to parentheses, as they can significantly affect the outcome of an expression.

In summary, in the absence of parentheses, negation affects only the base directly next to the negative sign, and not the result of the exponentiation.