Problem 21 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{14 x^{2}}{7 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{14x^{2}}{7x}\) is \(2x\).

Step 1: Simplify the Numerator and Denominator

Starting with the given expression, \(\frac{14x^{2}}{7x}\), the goal is to simplify the numerator and the denominator individually if possible. The numerator can be written as \(14*x*x\) and the denominator as \(7*x\). Here, the 14 in the numerator is divisible by the 7 in the denominator, which simplifies to 2. One 'x' in the numerator cancels out with the 'x' in the denominator.

Step 2: Simplify the Fraction

After simplifying, the fraction will reduce to \(\frac{2x}{1}\), whereby the '1' in the denominator does not need to be shown.

Step 3: Write Final Simplified Expression

The simplified version of the original fraction is \(2x\).

Key Concepts

Numerator and Denominator

In every rational expression, the numerator and the denominator play crucial roles. The numerator is the expression located above the fraction bar. It signifies the number of parts being discussed. Whereas, the denominator is the expression below the fraction bar. It points to the number of equal parts the whole is divided into.
In the given exercise, we start with the rational expression \(\frac{14x^2}{7x}\). The numerator is \(14x^2\), which can be considered as \(14 \times x \times x\), and the denominator is \(7x\). Recognizing these components is the first step toward simplifying the expression efficiently.
This breakdown allows us to identify common factors. Both the numerator and denominator share an 'x', and this needs to be factored out in order to simplify the expression further. Recognizing these factors is essential to finding the simplest form of the rational expression.

Cancelling Terms

Once the common factors between the numerator and the denominator are identified, it's time to cancel them. This is often referred to as cancelling terms, a vital step in simplifying rational expressions.
In our example, the term \(x\) appears in both the numerator and the denominator, which means they can cancel each other out. Similarly, the numbers 14 and 7 can be simplified by dividing them by their greatest common divisor, which is 7.
Here are a few steps to make cancelling terms easy:

• Look for common terms in both the numerator and the denominator.
• Identify the greatest common divisor for numerical coefficients.
• Divide both terms by the greatest common divisor.

This results in a cleaner expression, reducing our initial fraction \(\frac{14x^2}{7x}\) to a much simpler form. By removing these shared components, you'll end up with fewer and simpler terms, making calculations easier.

Algebraic Fractions

Algebraic fractions might sound complicated, but they are nothing more than fractions with polynomials in the numerator and the denominator. Simplifying these fractions often involves several key steps that help in reducing them to their simplest form.
The process begins with recognizing similar terms or factors in both parts of the fraction. Focusing on simplification, as we did in the expression \(\frac{14x^2}{7x}\), means factoring polynomials and cancelling out common terms. This leads us to the final simplified form \(2x\), achieved by straightforward multiplication and division among the terms involved.
Here are a few basic rules for handling algebraic fractions:

• Identify common factors in the numerator and the denominator.
• Use basic arithmetic to cancel these terms out.
• Ensure that you are dividing, multiplying, or simplifying each mathematical term correctly.

These practices help in dealing with algebraic fractions efficiently and provide a solid foundation for tackling more complex problems.