/*! 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 53 Write each rational expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 53

Write each rational expression in lowest terms. $$ \frac{a c-a d+b c-b d}{a c-a d-b c+b d} $$

Short Answer

Expert verified
The simplified expression is \( \frac{a + b}{a - b} \).

Step by step solution

01

Identify Common Factors in the Numerator

Examine the numerator \( ac - ad + bc - bd \). Notice that each pair of terms shares a common factor. Group the terms: \( (ac - ad) + (bc - bd) \). Factor out 'a' from the first group and 'b' from the second group: \( a(c - d) + b(c - d) \). Finally, factor out the common term \( c-d \): \( (c - d)(a + b) \).
02

Identify Common Factors in the Denominator

Similarly, examine the denominator \( ac - ad - bc + bd \). Group the terms: \( (ac - ad) - (bc - bd) \). Factor out 'a' from the first group and 'b' from the second group: \( a(c - d) - b(c - d) \). Factor out the common term \( c-d \): \( (c - d)(a - b) \).
03

Simplify the Expression

Combine the factored numerator and denominator: \( \frac{(c - d)(a + b)}{(c - d)(a - b)} \). Since \( c-d \) appears in both the numerator and the denominator, they cancel each other out. This reduces the expression to \( \frac{a + b}{a - b} \).

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 Polynomials
Polynomials are expressions that consist of variables and coefficients. They are combined using addition, subtraction, and multiplication. Factoring is the process of breaking down a polynomial into its simplest components or 'factors'. This is like splitting a number into its prime factors.
Numerator and Denominator
In a rational expression, the numerator is the expression above the fraction line, and the denominator is the expression below it. Think of the numerator and denominator as separate polynomials. When simplifying a rational expression, it is important to treat them individually first before combining.
Cancel Common Factors
When a factor appears in both the numerator and the denominator, it can be 'cancelled out'. This means it can be removed from the fraction. For example, in the expression \( \frac{(c - d)(a + b)}{(c - d)(a - b)} \), the \(c - d\) terms can cancel each other out. This is because any number divided by itself equals one.

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

Write each rational expression in lowest terms. $$ \frac{7 x-21}{63-21 x} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{3}{8} \div \frac{9}{14} $$

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

Add or subtract as indicated. $$\frac{6 x+5 y}{6 x^{2}+5 x y-4 y^{2}}-\frac{x+2 y}{9 x^{2}-16 y^{2}}$$

Add or subtract as indicated. $$\frac{r+s}{3 r^{2}+2 r s-s^{2}}-\frac{s-r}{6 r^{2}-5 r s+s^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.