/*! 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 Simplify the expression. $$5 \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 18

Simplify the expression. $$5 \sqrt[4]{9}+2 \sqrt[4]{a^{3}}-3 \sqrt[4]{9}+3 \sqrt[4]{a^{3}}$$

Short Answer

Expert verified
The simplified expression is \(2 \sqrt[4]{9} + 5 \sqrt[4]{a^{3}}\).

Step by step solution

01

Identify Like Terms

Find and highlight like terms in the expression. In this case, the like terms are \(5 \sqrt[4]{9}\) and \(-3 \sqrt[4]{9}\), as well as \(2 \sqrt[4]{a^{3}}\) and \(3 \sqrt[4]{a^{3}}\). By 'like terms' it is meant terms that contain the same variable to the same power.
02

Combine Like Terms

Next, combine like terms by addition or subtraction as indicated in the expression. This simplifies the expression to \(2 \sqrt[4]{9} + 5 \sqrt[4]{a^{3}}\). This is achieved by adding \(5 \sqrt[4]{9}\) and \(-3 \sqrt[4]{9}\) to get \(2 \sqrt[4]{9}\), and then adding \(2 \sqrt[4]{a^{3}}\) and \(3 \sqrt[4]{a^{3}}\) to get \(5 \sqrt[4]{a^{3}}\).
03

Simplify the Expression Further

Since no other simplifications are possible, the simplified expression remains as \(2 \sqrt[4]{9} + 5 \sqrt[4]{a^{3}}\).

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
Understanding like terms is crucial when simplifying algebraic expressions. Like terms are components of an algebraic expression that have the same variable raised to the same power, even if their coefficients (the numbers in front of the variables) differ. For example, in the expression 3x + 5x, both parts are like terms because they both contain the variable x to the first power.
Radical Expressions
A radical expression includes a root symbol with an index, indicating which root is being taken. In everyday mathematics, we often interact with square roots, but algebraic expressions can contain roots of any degree鈥攃ubed roots, fourth roots, and so on. For instance, the fourth root of the number 16 is 2, because 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 equals 16. When it comes to variables under a root, such as \(\sqrt[4]{a^{3}}\), we're dealing with higher complexity since the simplification depends on the degree of the root and the power of the variable.
Fourth Roots
When we talk about fourth roots, we refer to the operation of finding a number that, when multiplied by itself three more times (four times in total), gives you the original number. The expression for the fourth root of a number n is written as \(\sqrt[4]{n}\).
Algebraic Operations
Algebraic operations encompass actions such as addition, subtraction, multiplication, and division applied to algebraic expressions. The aim is often to simplify or rearrange the expression for easier understanding or further manipulation. Operating with algebraic expressions requires a keen eye on like terms, the recognization of radical expressions, and an understanding of specialized roots like fourth roots.

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

Number Sense Is the number \(3 /(1+\sqrt{5})\) in simplest form? If not, explain the steps for writing in simplest form.

Justifying Steps What properties or rules are used to rewrite \(\sqrt{2}(\sqrt{8}-\sqrt{2})\) as \(\sqrt{16}-\sqrt{4}\) in the solution above?

Find the conjugate of the expression. Then find the product of the expression and its conjugate. $$\sqrt{11}+\sqrt{5}$$

Factor the polynomial completely. $$x^{2}-25$$

Solve the equation. $$\sqrt{2 u-9}=-\sqrt{u}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.