91Ó°ÊÓ

Understanding the order of operations is fundamental in evaluating mathematical expressions, especially when they involve multiple steps and different operations. It is the hierarchy that math follows to ensure consistent results. The acronym PEMDAS is often used to help remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In evaluating the expression \( -10^{2} \), it's important to note that there are no parentheses to consider first. The next step is to address any exponents in the expression. Here, the expression inside the exponent is simply the base number 10 being squared, meaning 10 is multiplied by itself. Only after evaluating the exponential part do we apply the negative sign to the result since the negative is not within parentheses. Understanding this sequence helps avoid common mistakes, such as incorrectly applying the negative sign before the exponentiation.

When dealing with negative signs in exponential expressions, it's important to distinguish whether the negative applies to the base of the exponent or the entire expression. In the case of \( -10^{2} \), the negative sign is not part of the base; it is not \( (-10)^{2} \), which would yield a positive result due to the negative sign being squared.

Instead, the negative applies to the entire expression after evaluating the exponent, making the final result negative. If a negative sign is raised to an even exponent, the result is positive, while an odd exponent would yield a negative result. This concept reinforces why understanding the order of operations is crucial, as it impacts the interpretation and outcome of negative exponentiation.

Powers and exponents represent a shorthand way of expressing repeated multiplication. An exponent, placed to the upper right of a base number, indicates how many times to multiply the base by itself. For example, \(10^{2}\) is read as '10 squared', which means \(10 \times 10\).

It's important to recognize that exponents work differently than basic arithmetic operations. They involve their own set of rules, such as how to handle multiplication and division of expressions with exponents, raising a power to another power, and working with zero and negative exponents.

For the given expression \( -10^{2} \), you might be tempted to multiply \( -10 \times -10 \), but the exponent only applies to the base 10, not the negative sign. Thus, it demonstrates how exponents require careful attention to detail, especially when combined with other mathematical operations or signs.