Problem 69 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{-a}{-a^{2}-a}\)

Step-by-Step Solution

Answer

Question: Reduce the rational expression \(\frac{-a}{-a^{2}-a}\) to its lowest terms. Answer: \(\frac{1}{1 + a}\)

Step 1: 1. Simplify the numerator and denominator expressions

First, we need to simplify the numerator and denominator expressions: Numerator: \(-a\) Denominator: \(-a^{2} - a\)

Step 2: 2. Factor the expressions

Next, we need to factor both expressions. The numerator is already in its factored form. However, the denominator has a common factor of \(-a\) that we can factor out: Denominator: \(-a(1 + a)\)

Step 3: 3. Cancel out common factors

Now let's cancel out any common factors between the numerator and the denominator. We can see that both have a factor of \(-a\). Divide both by \(-a\): \(\frac{-a}{-a(1 + a)} \implies \frac{-a}{-a}\cdot\frac{1}{1 + a}\) After canceling \(-a\), we get: \(\frac{1}{1 + a}\) So, the given rational expression in its lowest terms is \(\frac{1}{1 + a}\).

Key Concepts

Factoring

Factoring is a vital technique in mathematics that helps simplify expressions. When it comes to rational expressions, factoring allows us to break down complex polynomials into simpler components, making it easier to perform operations such as addition, subtraction, and division. In the context of the given problem, factoring involves

• Identifying common terms in an expression,
• Extracting them to simplify the expression.

Consider the denominator of the given rational expression: \(-a^{2} - a\).Notice how both terms in the denominator share a common factor of \(-a\). By factoring out \(-a\), the expression simplifies from\(-a^{2} - a\) to \(-a(1 + a)\).This makes it much simpler. Factoring plays a crucial role in enabling further simplification of rational expressions, as it reveals common factors that can subsequently be canceled out.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest or lowest terms. For rational expressions, this means breaking down both the numerator and the denominator to identify and eliminate common factors. Simplification is a key step because a simpler form of an expression is easier to understand and work with.
In the example of the rational expression \(\frac{-a}{-a^2 - a}\), the simplification process starts by making sure each part of the fraction is factored as much as possible.

• The numerator, \(-a\), is already as simple as it can be.
• The denominator is factored to \(-a(1 + a)\), which reveals its components.

Successfully simplifying the rational expression requires recognizing these forms to cancel out common parts, leading to the simplest possible representation.

Canceling Common Factors

Canceling common factors is a powerful process in simplifying rational expressions. By identifying terms that appear in both the numerator and the denominator, it allows for reduction to the simplest form. This process involves recognizing common factors and dividing them out of both the numerator and the denominator.
In our exercise, the rational expression is \(\frac{-a}{-a(1+a)}\). Here, we can see that \(-a\) appears in both the numerator and the denominator, making it a common factor. To cancel these common factors:

• Divide both the numerator and the denominator by \(-a\).

This leaves us with \(\frac{1}{1+a}\) since \(-a\) cancels itself out. Canceling common factors simplifies the expression to its lowest terms, making complex calculations or further algebraic processes much easier to manage.