Problem 15 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{12 x}{6 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{x} \).

Step 1: Identify common factors in the numerator and denominator

The numerator is 12x and the denominator is 6x². We observe that both 12 and 6 can be divided by their greatest common divisor, which is 6. The variable x in the numerator and x² in the denominator indicates that the common factor here is x.

Step 2: Simplify by dividing both terms by their common factors

Divide 12 by 6 to get 2, and divide x by x to get 1 in the numerator. Similarly, divide 6 by 6 to get 1 and x² by x to get x in the denominator. After canceling out the common factors, the simplified expression is \( \frac{2}{x} \).

Key Concepts

Common Factors

Common factors are numbers or variables that appear in both the numerator and the denominator of a fraction. Identifying these is the first step in simplifying rational expressions.
Let's take the expression \( \frac{12x}{6x^2} \) as an example. In this case, both the numerator, 12, and the denominator, 6, can be factored into smaller numbers:

• 12 can be factored into 2 and 6.
• 6 can be factored as 6 itself.

The common numerical factor is 6. Additionally, we have a variable \( x \) in both terms. Hence, \( x \) is a common factor.
Recognizing these factors is crucial for the next step: simplification.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. It ensures the simplification process is as efficient as possible.
For instance, in \( \frac{12x}{6x^2} \), we identify the GCD of the numbers 12 and 6. This is 6, because it is the largest number that both numbers can be divided by evenly:

• 12 divided by 6 equals 2.
• 6 divided by 6 equals 1.

Finding the GCD helps us know how to reduce the terms effectively. It streamlines the process of simplification, leading us toward the simplest form of the expression.

Numerator and Denominator

In any fraction, the numerator is the top part, and the denominator is the bottom part. Understanding their roles is key in manipulating rational expressions.
For \( \frac{12x}{6x^2} \):

• The **numerator** is 12x. It combines both a number and a variable.
• The **denominator** is 6x², which includes a number and a squared variable.

Both parts need attention when simplifying expressions. We look for common factors in both to rewrite the fraction in a simpler form. This understanding aids in rearranging the fraction visibly to see which parts can be eliminated.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves dividing each term by their common factors, thus condensing it into a simpler form.
Starting with the expression \( \frac{12x}{6x^2} \):

• Divide the numerical part: 12 divided by 6 equals 2.
• Divide variable parts: \( x \) by \( x^2 \) results in \( x^{1-2} = \frac{1}{x} \).

This results in a simplified expression: \( \frac{2}{x} \).
Simplification not only makes expressions easier to work with, but it also reveals the simplest relation between components of the original expression, making calculations and further operations more manageable.