Problem 13 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{6}{6 x-18} $$

Step-by-Step Solution

Answer

Question: Reduce the given rational expression to its lowest terms: $$\frac{6}{6x-18}$$ Answer: $$\frac{1}{x-3}$$

Step 1: Identify the common factor

In this rational expression, we can see that both the numerator (6) and the denominator (6x - 18) share a common factor, which is 6.

Step 2: Factor out the common factor

Now we will factor out the common factor of 6 from both the numerator and the denominator. This will give us: $$ \frac{6}{6x-18} = \frac{6}{6(x-3)} $$

Step 3: Cancel out the common factor

Finally, we will cancel out the common factor by dividing both numerator and denominator by 6: $$ \frac{6}{6(x-3)} = \frac{6\div 6}{6\div 6 (x-3)} = \frac{1}{x-3} $$

Step 4: Write the final answer

The rational expression has been reduced to its lowest terms: $$ \frac{6}{6x-18} = \frac{1}{x-3} $$

Key Concepts

Common Factor

Understanding the concept of the common factor is crucial when simplifying rational expressions. A common factor is a number or algebraic term that divides exactly into two or more numbers or terms. For instance, in the given exercise, we are looking at the rational expression \( \frac{6}{6x-18} \). By examining the numerator (6) and the denominator (6x - 18), we identify that 6 is a common factor since it can divide both terms without leaving a remainder.

Recognizing common factors allows us to simplify expressions and solve equations more efficiently. Always look for numbers and variables that appear in each part of the expression as this can dramatically reduce its complexity.

Reduce to Lowest Terms

The ultimate goal when working with rational expressions is to reduce them to their lowest terms. This means simplifying the expression so that the numerator and denominator are as simple as possible and have no common factors other than 1. Reducing an expression to its lowest terms makes it more comprehensible and easier to work with in subsequent calculations.

To achieve this, after you've identified the common factor, you divide both the numerator and denominator by this factor. As demonstrated in the solution, dividing both terms by 6 gave us a simpler and more elegant expression, \( \frac{1}{x-3} \). By simplifying a complex expression to its lowest terms, we can often make the relationships between variables and constants clearer and the solution more straightforward.

Factor Out

The process of 'factoring out' refers to taking a common factor from terms and writing it outside of an expression's main body. It is a powerful tool for simplifying equations and expressions. In the provided exercise, we used this process to factor out the number 6 from the denominator \(6x-18\), resulting in the expression \(6(x-3)\).

Factoring out not only makes an expression neater and more organized but also sets the stage for easily canceling out common factors. It's similar to decluttering: by removing the redundancies (the common factor), you're left with the essence of the expression. Remember, when you factor something out, you're not eliminating it; you're simply reorganizing the expression to prepare it for the next step, which usually involves canceling out or further simplification.