Problem 51 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{a+7}{a-1}\)

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is \(\frac{a+7}{a-1}\).

Step 1: Identify the numerator and the denominator

The expression is a fraction, where the numerator is \(a+7\) and the denominator is \(a-1\).

Step 2: Check for common factors

In the given expression, there are no common factors between the numerator and the denominator.

Step 3: Determine if the expression can be further simplified

Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further.

Step 4: Write the final answer

As the expression cannot be simplified, the answer in its lowest terms is: \(\frac{a+7}{a-1}\).

Key Concepts

Understanding Numerator and Denominator

In a rational expression, we deal with two important components: the numerator and the denominator. The numerator is the top part of the fraction, which in this case is \(a + 7\), while the denominator is the bottom part, \(a - 1\). These components form the basis of any fraction. It's essential to recognize these parts because it's the first step in most rational expression exercises. Identifying them correctly helps us focus on what needs to be simplified, and it sets the groundwork for understanding the relationship between the two. When tackling problems involving fractions or rational expressions, always start by identifying your numerator and denominator.

The Simplification Process

Simplifying a rational expression is all about reducing it to its simplest form. This involves looking for any possible reduction in terms between the numerator and the denominator. The goal is to remove any complexity from the expression. A simplified expression is easier to interpret and manage, especially when performing further mathematical operations.To decide if an expression like \(\frac{a+7}{a-1}\) can be simplified, we must check if any terms or factors are shared between the numerator and the denominator. If there are shared factors, we can cancel them out, simplifying the expression. However, in this exercise, the terms \(a+7\) and \(a-1\) share no common factors, meaning this rational expression is already in its simplest form.

Importance of Common Factors

Common factors play a crucial role in simplifying rational expressions. They are the elements that appear in both the numerator and the denominator, allowing us to reduce the expression by cancelling these common elements. The presence of common factors means there is redundancy that can be eliminated, which streamlines the expression.For the expression \(\frac{a+7}{a-1}\), we need to determine if a common factor exists. In this case, neither the terms nor the coefficients, apart from the variable itself, are common between the numerator and the denominator. This lack of common factors indicates that no further simplification is possible. When approaching any rational expression, always inspect for common factors as your primary step towards simplification.