/*! 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 82 Perform the operations. $$ \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 82

Perform the operations. $$ \left(5.7 n^{3}-2.1 n\right)+\left(-6.2 n^{3}-3.9 n\right) $$

Short Answer

Expert verified
The simplified expression is \(-0.5n^3 - 6.0n\).

Step by step solution

01

Identify Like Terms

Before performing the addition, identify the like terms in the given expression. Like terms are terms that have the same variable raised to the same power. Here, we have the terms with \(n^3\) and the terms with \(n\).
02

Group Like Terms Together

Group the coefficients of like terms in the expression:\[(5.7n^3 - 6.2n^3) + (-2.1n - 3.9n)\]
03

Add/Subtract Like Terms

Perform the operations for each group of like terms:1. Combine the \(n^3\) terms: \[5.7n^3 - 6.2n^3 = -0.5n^3\]2. Combine the \(n\) terms: \[-2.1n - 3.9n = -6.0n\]
04

Write the Final Expression

Combine the results to write the final simplified expression:\[-0.5n^3 - 6.0n\]

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.

Like Terms
When working with polynomials, the concept of like terms is foundational. Like terms are terms that share the same variable and the same exponent. For example, in an expression like \(5.7n^3 - 6.2n^3\) and \(-2.1n - 3.9n\), the terms \(5.7n^3\) and \(-6.2n^3\) are like terms because they both contain \(n^3\). Similarly, \(-2.1n\) and \(-3.9n\) are like terms as both contain \(n\).
Identifying like terms is crucial because it tells us which terms can be combined. Terms that do not share the exact variable and exponent cannot be combined directly. This process simplifies complex polynomial expressions, helping solve and understand them better.
Combining Terms
Once you have identified the like terms, the next step is to combine them. This process involves adding or subtracting the coefficients (the numerical parts) of these terms.
Consider the expression \(5.7n^3 - 6.2n^3\). The like terms here have the same variable \(n^3\) and exponent 3. To combine them:
  • Subtract the smaller coefficient \(6.2\) from the larger \(5.7\)
  • This results in \(-0.5n^3\)
Similarly, for \(-2.1n - 3.9n\), combine their coefficients to obtain \(-6.0n\).
This process is about simplifying the expression by performing basic arithmetic operations on the coefficients of like terms.
Simplifying Expressions
Simplifying expressions is an essential skill in algebra, allowing us to rewrite expressions in their simplest form. This process often involves several steps:
First, identify and group the like terms, as seen in polynomials. Next, combine the coefficients of these like terms. Finally, rewrite the expression.
In our example, we started with the terms \(5.7n^3 - 6.2n^3\) and \(-2.1n - 3.9n\). After combining like terms, the expression simplified to \(-0.5n^3 - 6.0n\).
  • This step helps in reducing complex expressions into a form that is easier to handle or solve.
  • Simplifying makes it simpler to evaluate or substitute numbers for variables efficiently.
Being comfortable with simplifying expressions is essential for tackling more advanced problems in algebra.

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

Writing \((x+y)^{2}\) as \(x^{2}+y^{2}\) illustrates a common error. Explain.

Perform each division. $$ \left(x^{2}+6 x+15\right) \div(x+5) $$

Perform each division. $$ \frac{5 x^{4}-10 x}{25 x^{3}} $$

Perform the indicated operations. a. \((x y)^{2}\) b. \((x+y)^{2}\)

Perform the indicated operations to simplify each expression, if possible. a. \(3 a+(4 a-1)+(6 a+2)\) b. \(3 a(4 a-1)(6 a+2)\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.