/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Write each rational expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 54

Write each rational expression in lowest terms. $$ \frac{2 x y+2 x w+y+w}{2 x y+y-2 x w-w} $$

Short Answer

Expert verified
\[= \frac{y + w}{y - w}\]

Step by step solution

01

Factor out common terms

First, identify and factor out common terms from the numerator and the denominator. The expression is:\[\frac{2 x y + 2 x w + y + w}{2 x y + y - 2 x w - w}\]In the numerator, factor by grouping:\[= \frac{2 x (y + w) + 1 (y + w)}{2 x y + y - 2 x w - w}\]So the numerator can be factored as:\[= \frac{(2 x + 1)(y + w)}{2 x y + y - 2 x w - w}\]
02

Factor the denominator

Similarly, factor by grouping in the denominator:\[= \frac{(2 x + 1)(y + w)}{2 x y - 2 x w + y - w}\]The denominator simplifies to:\[= \frac{(2 x + 1)(y + w)}{2 x (y - w) + 1 (y - w)}\]So the denominator is factored as:\[= \frac{(2 x + 1)(y + w)}{(2 x + 1)(y - w)}\]
03

Cancel out common factors

Now, cancel out the common factors in the numerator and the denominator:\[= \frac{(2 x + 1)(y + w)}{(2 x + 1)(y - w)}\]\[= \frac{y + w}{y - w}\]So the simplified expression is:\[= \frac{y + w}{y - w}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Factoring by Grouping
Factoring by grouping is a method used to simplify polynomials, especially when dealing with four terms. This technique involves arranging and combining terms into pairs that can be factored easily.
For instance, in the numerator \[2xy + 2xw + y + w\], we can group \[2xy + 2xw\] and \[y + w\].
Now, factor out the common terms within each group. From \[2xy + 2xw\], factor out \[2x\], giving us \[2x(y + w)\]. In the second group, \[y + w\] is common, so rewrite it as \[1(y + w)\].
This grouping lets us rewrite the whole numerator as \[2x(y+w) + 1(y+w)\], further simplifying to \[(2x+1)(y+w)\].
Similarly, apply this method to the denominator.
Common Factors
Finding common factors is crucial for simplifying expressions. For the denominators, you deal with \[2xy + y - 2xw - w\].
Using the same grouping strategy, group terms: \[2xy - 2xw \] and \[y - w\].
In the first group, \[2x(y - w)\] is factored out. In the second group, factor \[1(y - w)\]. This lets us write the denominator as \[2x(y - w) + 1(y - w)\], simplifying to \[(2x + 1)(y - w)\].
We now have factored forms for both the numerator and the denominator, making the next step straightforward.
Lowest Terms
Simplifying to lowest terms means canceling out any common factors between the numerator and the denominator. In our factored rational expression, we have:
  • Numerator: \[(2x + 1)(y + w)\]
  • Denominator: \[(2x + 1)(y - w)\]
Here, \[2x + 1\] is a common factor.
We can cancel it out from both the numerator and the denominator.
After canceling \[2x + 1\], we are left with \[\frac{y + w}{y - w}\]. This fraction is in its lowest terms, meaning it cannot be simplified further.
Thus, our final simplified expression is \[\frac{y + w}{y - w}\].

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Add or subtract as indicated. $$\frac{3}{(p-2)^{2}}-\frac{5}{p-2}+4$$

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

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{2}{3} \div \frac{8}{9} $$

Write each rational expression in lowest terms. $$ \frac{t^{2}-9}{3 t+9} $$

Write each rational expression in lowest terms. $$ \frac{2 x^{2}-5 x}{16 x-40} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.