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Chapter 7: Problem 33

Write each rational expression in lowest terms. $$ \frac{6 m+18}{7 m+21} $$

Short Answer

Expert verified
The simplified rational expression is \( \frac{6}{7} \).

Step by step solution

01

Identify Common Factors in the Numerator

The numerator of the expression is \(6m + 18\). First, identify the common factors in each term of the numerator. Both terms share a common factor of 6.
02

Factor the Numerator

Factor out the common factor of 6 from the numerator: \[ 6m + 18 = 6(m + 3) \]
03

Identify Common Factors in the Denominator

Next, look at the denominator of the expression, which is \(7m + 21\). Both terms share a common factor of 7.
04

Factor the Denominator

Factor out the common factor of 7 from the denominator: \[ 7m + 21 = 7(m + 3) \]
05

Simplify the Rational Expression

Now we have the expression in factored form: \[ \frac{6(m+3)}{7(m+3)} \] Since \(m + 3\) is a common factor in both the numerator and the denominator, they cancel each other out: \[ \frac{6(m + 3)}{7(m + 3)} = \frac{6}{7} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
When simplifying rational expressions, the first vital step is factoring polynomials. Polynomials are algebraic expressions consisting of variables and coefficients. For example, in the expression \(6m + 18\), we want to factor it out.

To factor a polynomial, look for the greatest common factor (GCF). Here, 6 is the GCF of both 6m and 18.

By factoring out the common factor, we transform \(6m + 18\) into \(6(m + 3)\). This makes simplifying the expression easier in later steps.
Common Factors
Identifying common factors plays a crucial role in simplifying rational expressions. A common factor is a number or variable that evenly divides every term in the expression.

Taking \(7m + 21\) as an example, the number 7 is the common factor. When we identify 7 as the GCF, we can factor it out from the expression: \(7m + 21 = 7(m + 3)\).

This simplifies the process of cancelling terms in the rational expression.
Cancelling Terms
Once we have factored both the numerator and the denominator, the next step is to cancel common terms in the rational expression.

In our factored expression \(\frac{6(m+3)}{7(m+3)}\), the \(m + 3\) term appears in both the numerator and the denominator.

Since \(m + 3\) is a common term, we can cancel these identical terms. This results in a simplified expression: \(\frac{6}{7}\).
Numerator and Denominator
To fully grasp the simplification of rational expressions, understanding the roles of the numerator and denominator is crucial. The numerator is the top part of the fraction, and the denominator is the bottom part.

In our given expression \(\frac{6m + 18}{7m + 21}\), the numerator is \(6m + 18\), and the denominator is \(7m + 21\).

By factoring both the numerator and the denominator, and cancelling common terms, we transform a complex expression into its simplest form: \(\frac{6}{7}\).

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Most popular questions from this chapter

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