/*! 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 41 Write each rational expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 41

Write each rational expression in lowest terms. $$ \frac{m^{2}-n^{2}}{m+n} $$

Short Answer

Expert verified
m - n

Step by step solution

01

Identify the Numerator and Denominator

The numerator is \( m^2 - n^2 \) and the denominator is \( m + n \).
02

Factor the Numerator

Notice that the numerator \( m^2 - n^2 \) is a difference of squares, which can be factored as \( (m-n)(m+n) \).
03

Write the Factored Form

After factoring the numerator, the expression becomes \( \frac{(m-n)(m+n)}{m+n} \).
04

Simplify the Fraction

Since \( m+n \) appears in both the numerator and the denominator, they can be cancelled out. Therefore, the simplified expression is \( m - n \).

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 differences of squares
One important concept in algebra is factoring the difference of squares. This method can help simplify complex expressions.
An expression like \( m^2 - n^2 \) fits this form. How do we recognize this pattern?
When you see two squared terms separated by a minus sign, it鈥檚 a difference of squares.
To factor it, recall this formula: \[ a^2 - b^2 = (a + b)(a - b) \]
In our exercise, \( a = m \) and \( b = n \). Thus, \( m^2 - n^2 = (m + n)(m - n) \).
This simplification will help us reduce the overall expression in later steps.
numerator and denominator
Next, it鈥檚 crucial to understand the roles of the numerator and denominator in a fraction.
The **numerator** is the top part of the fraction. In our example, it's \( m^2 - n^2 \). The **denominator** is the bottom part. In this case, it鈥檚 \( m + n \).
When simplifying rational expressions, always look at these parts independently at first. This will help identify any common factors or special patterns, like the difference of squares mentioned earlier. By isolating each part, you can more easily see how to proceed with simplification.
simplification of fractions
Simplifying fractions involves reducing them to their lowest terms. This means that there are no common factors in the numerator and denominator other than 1. After factoring the numerator and observing the denominator, we noticed they both had \( (m+n) \). When the numerator and denominator share a common factor, it can be canceled out.
In our example: \[ \frac{(m - n)(m + n)}{m + n} \]
We can cancel \( (m + n) \) from both the top and bottom, leaving us with \( m - n \). It's important to remember this rule: You can cancel terms only when multiplying, not adding or subtracting. This step is essential for simplifying expressions efficiently.

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

Multiply or divide. Write each answer in lowest terms. $$ \frac{(q-3)^{4}(q+2)}{q^{2}+3 q+2} \div \frac{q^{2}-6 q+9}{q^{2}+4 q+4} $$

Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{2}{3} \cdot \frac{5}{6} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{6}{k^{2}-4 k}=\frac{?}{k(k-4)(k+1)} $$

Multiply. Write each answer in lowest terms. $$ \frac{12 x^{4}}{18 x^{3}} \cdot \frac{-8 x^{5}}{4 x^{2}} $$

Divide. Write each answer in lowest terms. $$ \frac{y^{2}}{y+1} \div \frac{3 y}{y-3} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.