/*! 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 120 How are the procedures used to s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 120

How are the procedures used to simplify \(3 x+4 x\) and \(3 \sqrt{x}+4 \sqrt{x}\) similar?

Short Answer

Expert verified
Both procedures involve factoring out a common term and simplifying the coefficient.

Step by step solution

01

Observe Common Terms

For both expressions, identify the common terms found within each component. In the expression \(3x + 4x\), both terms include the variable \(x\). Likewise, in the expression \(3\sqrt{x} + 4\sqrt{x}\), both terms include \(\sqrt{x}\) as the common factor.
02

Factor Out the Common Term

In the expression \(3x + 4x\), factor out the common variable \(x\). You will get \((3 + 4)x\). Similarly, in the expression \(3\sqrt{x} + 4\sqrt{x}\), factor out the common term \(\sqrt{x}\) to obtain \((3 + 4)\sqrt{x}\).
03

Simplify the Sum Inside Parentheses

Calculate the arithmetic inside the parentheses for both expressions. Compute \(3 + 4 = 7\). Therefore, both expressions simplify to \(7x\) and \(7\sqrt{x}\) respectively.

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.

Simplifying Expressions
Simplifying expressions is a fundamental process in algebra that means breaking down complex expressions into their simplest form. This is often done by combining **like terms** or **factoring** out common factors.

To simplify an expression, you first identify terms within the expression that can be simplified, such as like terms (terms that have the same variables raised to the same power). Once these terms are recognized, you can perform operations to combine or reduce them into fewer terms.
  • For instance, in the algebraic expression \(3x + 4x\), the terms are like terms because both have the variable \(x\).
  • You can add the coefficients (numerical factors) of these terms together to simplify the expression to \(7x\).
This process helps to make expressions easier to work with, especially when you need to perform further algebraic manipulations such as solving equations.
Factoring
Factoring plays a pivotal role in simplifying algebraic expressions and equations. It involves breaking down an expression into products of simpler or more fundamental expressions. This is akin to "unwrapping" the expression into factors that when multiplied together give the original expression.

In the simplification of expressions like \(3x + 4x\) and \(3\sqrt{x} + 4\sqrt{x}\), one common method involves **factoring out** the greatest common factor (GCF).
  • You look for the GCF of the terms, which is \(x\) for the first expression and \(\sqrt{x}\) for the second.
  • By factoring this out, you rewrite \(3x + 4x\) as \((3 + 4)x\) and \(3\sqrt{x} + 4\sqrt{x}\) as \((3 + 4)\sqrt{x}\).
Whether dealing with numbers, variables, or combinations, factoring is a key technique that greatly simplifies expressions and is widely used in solving equations.
Like Terms
Identifying and combining **like terms** is critical in the simplification process for algebraic expressions. Like terms are those that have identical variables raised to the same power, even though they might have different coefficients.

When you encounter like terms, adding or subtracting them is straightforward and can lead to a much simpler expression.
  • Consider \(3x + 4x\); both terms are like terms because they contain the variable \(x\).
  • Similarly, in \(3\sqrt{x} + 4\sqrt{x}\), both terms are like terms sharing \(\sqrt{x}\).
By adding the coefficients of these terms, you simplify them down to fewer terms: \(7x\) and \(7\sqrt{x}\), respectively. This not only reduces complexity but also makes further mathematics operations easier to perform.

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

Simplify each expression. All variables represent positive real numbers. $$ 16^{-3 / 2} $$

Simplify each expression. All variables represent positive real numbers. $$ \sqrt[4]{25 b^{2}} $$

Simplify each expression. All variables represent positive real numbers. $$ \left(25 y^{2}\right)^{1 / 2} $$

Simplify each expression. All variables represent positive real numbers. a. \(81^{1 / 4}\) b. \(81^{-1 / 4}\) c. \(-81^{1 / 4}\) d. \(\frac{1}{81^{-1 / 4}}\)

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{-216 z^{9}} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.