/*! 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 18 Multiply and, if possible, simpl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 18

Multiply and, if possible, simplify. $$\frac{m^{2}-n^{2}}{4 m+4 n} \cdot \frac{m+n}{m-n}$$

Short Answer

Expert verified
\( \frac{m + n}{4} \)

Step by step solution

01

Factor the numerator of the first fraction

The numerator of the first fraction is a difference of squares: \( m^2 - n^2 = (m + n)(m - n) \)
02

Simplify the first fraction

Substitute the factored form into the first fraction: \( \frac{(m + n)(m - n)}{4(m + n)} \)
03

Cancel common factors in the first fraction

Cancel out the common factor \( m + n \) in the numerator and the denominator: \( \frac{m - n}{4} \)
04

Write the simplified product

Multiply the simplified first fraction by the second fraction: \( \frac{m - n}{4} \cdot \frac{m + n}{m - n} \)
05

Cancel common factors in the product

Cancel out the common factor \( m - n \): \( \frac{m + n}{4} \)

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.

difference of squares
In algebra, recognizing the difference of squares can simplify many problems. The difference of squares formula lets you break down a quadratic expression into two binomials. The general form is: \[a^2 - b^2 = (a - b)(a + b)\]In our exercise, we used this method on \(m^2 - n^2\). By identifying it as a difference of squares, it became:\[m^2 - n^2 = (m + n)(m - n)\]This step simplifies further computations, making the whole problem more manageable. Notice how both binomials involve the same variables, but with different signs; this is the crux of the method.
factoring
Factoring means expressing an expression as a product of its factors. In our problem, after recognizing the difference of squares, we then dealt with the factorization directly:\[\frac{(m + n)(m - n)}{4m + 4n}\]Next, we simplified the original complex fraction into simpler terms. This involved factoring out common terms in the numerator and the denominator. Note that if there are terms like 4\((m+n)\) in the denominator, you should express them clearly to see what's cancellable. Factoring is essential in simplifying algebraic expressions and making them easier to handle.
simplification
Simplification involves reducing fractions or algebraic expressions to their simplest form. In our exercise, after factoring, we then simplified the expression:\[\frac{(m+n)(m-n)}{4(m+n)}\]Cancelling the common factor \(m+n\) in the numerator and denominator, we got:\[\frac{m-n}{4}\]This fraction was then multiplied with \(\frac{m+n}{m-n}\), leading us to cancel \(m-n\), resulting in the simplified form:\[\frac{m+n}{4}\]Simplification helps to reduce expressions to a form that's easier to work with and understand. It is crucial in solving algebraic fractions 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

The volume \(V\) of a given mass of a gas varies directly as the temperature \(T\) and inversely as the pressure \(P .\) If \(V=231 \mathrm{cm}^{3}\) when \(T=300^{\circ} \mathrm{K}\) (Kelvin) and \(P=20 \mathrm{lb} / \mathrm{cm}^{2},\) what is the volume when \(T=320^{\circ} \mathrm{K}\) and \(P=16 \mathrm{Ib} / \mathrm{cm}^{2} ?\)

Find the LCM. $$2(y-3), 6(y-3)$$

Multiply and simplify. \(-1(2 x-7)\)

If the LCM of two numbers is their product, what can you conclude about the two numbers?

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\). $$f(x)=\frac{x+3}{x+2}-\frac{x+4}{x+3}, g(x)=\frac{x+5}{x+4}-\frac{x+6}{x+5}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

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.