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When solving mathematical expressions, it's essential to follow the order of operations. This ensures that everyone gets the same result when solving the same problem. The order is often remembered by the acronym 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 \( \frac{-3^2}{(-3)^2} \), we start by addressing any parentheses first. Then we move to calculate the exponents before diving into multiplication, division, addition or subtraction. In our simplified expression, we initially handle the exponentiation part before resolving the rest. This step-by-step approach is crucial in ensuring accuracy in more complex problems.

Exponentiation is a mathematical operation, written as \(a^b\), involving two numbers, the base \(a\) and the exponent \(b\). The exponent refers to the number of times the base is multiplied by itself. For instance, \(3^2\) means \(3 \times 3 = 9\).
Applied to our problem, we have two different exponentiations: \(-3^2\) and \((-3)^2\). It's important to know the distinction.
For \(-3^2\), the exponent applies to the 3 only, giving us -9 because of the negative sign outside the exponent. However, when evaluating \((-3)^2\), the exponent applies to everything inside the parentheses, resulting in 9 because \((-3) \times (-3) = 9\). This careful observation of where the exponent applies can make a big difference in the final outcome of a mathematical problem.

In a fraction, the top number is called the numerator, and the bottom number is called the denominator. These terms are essential for understanding and working with fractions.
The numerator represents how many parts of the whole we are considering. In the fraction \( \frac{-3^2}{(-3)^2} \), the numerator is \(-3^2\) which simplifies to \(-9\).
The denominator shows into how many parts the whole is divided. Here, \((-3)^2\) simplifies to 9.
When simplifying fractions, we perform operations separately on the numerator and the denominator before combining them. By dividing the simplified numerator (-9) by the simplified denominator (9), our result is \(-1\).
Understanding the roles of the numerator and denominator is crucial for fraction operations like simplification, addition, subtraction, multiplication, and division.