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Chapter 6: Problem 56

Write each rational expression in lowest terms. $$ \frac{r-13}{13-r} $$

Short Answer

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Step by step solution

01

Identify the Rational Expression

The given rational expression is \( \frac{r-13}{13-r} \).
02

Factor the Numerator and Denominator

Observe that the denominator can be written as a multiple of the numerator with a factor of -1. Rewrite it: \( 13-r = -(r-13) \).
03

Simplify the Expression

Using the factored form of the denominator, the expression becomes \( \frac{r-13}{-(r-13)} \). By cancelling the common term \(r-13\), we get: \( \frac{r-13}{-(r-13)} = -1 \).

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Key Concepts

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

rational expressions
A rational expression is a fraction in which the numerator and the denominator are both polynomials. For example, in this exercise, we have the rational expression \[ \frac{r-13}{13-r} \].
Understanding rational expressions lay the foundation for many algebraic manipulations. Rational expressions can look complicated, but they become much simpler when you know how to work with them.
They often require steps like factoring or simplifying to put them into a more familiar or useful form.
It's crucial to identify common factors and other patterns to make these simplifications easier.
factoring
Factoring is a key step in simplifying rational expressions. In this exercise, we want to factor the numerator and the denominator to find common terms. When we look at the expression \( \frac{r-13}{13-r} \), we see that the denominator can be rewritten.
This matching of terms is the essence of factoring. Here, we identify that \(13-r\) can be written as \(-(r-13)\), transforming the expression into \( \frac{r-13}{-(r-13)} \).
By rewriting the terms in this manner, simplification becomes possible. Hence, factoring is about recognizing patterns and rewriting them in a more convenient way.
simplification
Simplification involves reducing expressions to their simplest form. After factoring both the numerator and the denominator in our rational expression \( \frac{r-13}{-(r-13)} \), we see that \(r-13\) appears in both.
By canceling these common terms, we simplify the expression to \(-1\).
Each time you simplify, make sure you accurately perform the operations and fully understand why terms can be cancelled. Simplifying an expression makes it easier to work with in later calculations or for finding further solutions.
lowest terms
Writing a rational expression in its lowest terms means reducing it so that the numerator and the denominator have no common factors other than 1.
In our problem, after factoring and cancelling common terms in \( \frac{r-13}{-(r-13)} \), we end up with \(-1\).
This is a simple form and the lowest term, as it cannot be reduced any further. It's important to simplify and put expressions in their lowest terms to better understand and work with them in various mathematical problems.
This process ensures clarity and precision in your algebraic solutions.

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Most popular questions from this chapter

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{5}{9} \cdot \frac{12}{25} $$

Multiply or divide as indicated. $$ \frac{12 x-20}{5 x} \cdot \frac{6}{9 x-15} $$

Write each rational expression in lowest terms. $$ \frac{2 t+6}{t^{2}-9} $$

Multiply or divide as indicated. $$ \frac{p^{3}-27 q^{3}}{p^{2}+p q-12 q^{2}} \cdot \frac{p^{2}-2 p q-24 q^{2}}{p^{2}-5 p q-6 q^{2}} $$

Add or subtract as indicated. $$\frac{15}{y^{2}+3 y}+\frac{2}{y}+\frac{5}{y+3}$$
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