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Rational expressions are like fractions, but instead of just numbers in the numerator and denominator, they can contain variables too. Just as we simplify numerical fractions by eliminating common factors in the numerator and denominator, we also simplify rational expressions in algebra. Simplifying can involve factoring polynomials and canceling out factors that appear in both the numerator and denominator. It’s important to note, however, that variables in rational expressions can take on certain values that could make the expression undefined - usually, when they would cause division by zero. Hence, the domain of the expression (the values the variables are allowed to take) often needs to be considered.

Mathematical operations with numbers are foundational to algebra. Numerical operations include addition, subtraction, multiplication, division, and sometimes more advanced operations such as exponentiation. Each operation has properties and rules governing it, such as the distributive property for multiplication over addition or subtraction. Understanding these operations with numbers is essential before tackling more complex tasks like manipulating algebraic expressions or solving equations.

Simplifying a fraction means to make it as simple as possible, which usually involves making the numerator and the denominator as small as possible. This is done by finding the greatest common divisor (GCD) of both numbers and dividing both by it. Simplifying fractions reduces complexity and can make other operations easier to carry out. Calculating the GCD might involve prime factorization or other methods, but the key is to identify and divide out any common factors. In algebra, this concept extends to simplifying rational expressions, where we simplify by canceling out common algebraic factors.

When dividing negative numbers, remember that two negatives make a positive. If both the numerator and the denominator of a fraction are negative, the fraction itself will simplify to a positive number. This is because a negative divided by a negative results in a positive value. Also, a negative divided by a positive (or vice versa) results in a negative outcome. This fundamental idea helps us when working with rational expressions, particularly when simplifying them to their simplest form.