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When solving mathematical problems, it's essential to perform operations in the correct sequence to avoid errors. The order of operations, often remembered using the acronym PEMDAS or BODMAS, stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule guides us on which steps to tackle first in a complex expression.

For instance, consider our example problem \( \frac{-18}{-2(3)} \). According to the order of operations, we handle the multiplication within the parentheses first. So, we calculate \( -2 \times 3 \) before addressing the division by \( -18 \) on the numerator, ensuring that the fraction is simplified correctly. Once the parentheses have been taken care of, we can proceed to the next steps without confusion.

Dividing negative numbers might seem tricky, but it follows a simple rule: a negative divided by a negative gives a positive result, while a negative divided by a positive (or vice versa) results in a negative number. This holds true because the concept of division is tightly linked with multiplication, and the negative signs effectively cancel each other out.

Looking back at our problem, \( -18/-6 \), we see that the numerator and the denominator are both negative. According to the rule, this means the fraction simplifies to a positive number. Hence, we divide 18 by 6 to get 3. Understanding how division of negatives works helps to make sense of why the simplified fraction results in a positive number, contributing to the student's deeper grasp of negative number arithmetic.

Simplifying fractions is a process of reducing them to their simplest form – that is, to where the numerator and the denominator have no common factors other than 1. To do this efficiently, one should first factor the numerator and denominator if possible, and then divide both by their greatest common divisor.

In our example, once the denominator is simplified to \( -6 \), we then look at the fraction \( \frac{-18}{-6} \). Since both numbers are multiples of 6, we can divide them by 6 to simplify the fraction. The simplification process reveals that there are no common factors remaining other than 1. Therefore, the fraction \( \frac{-18}{-6} \) simplifies to \( 3 \), showcasing a clear understanding of fraction simplification and how to apply it to algebraic fractions containing negative numbers.