/*! 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 6 Given the expression: \((((a-b)+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Given the expression: \((((a-b)+b)+b)(a-b)+b^{2}\) (a) Simplify the expression using the associative law ONLY. (b) Simplify the expression using the associative and distributive laws ONLY. (c) Simplify the expression using the associative, distributive, and commutative laws.

Short Answer

Expert verified
Resultant expression: Simplifyinga using different laws

Step by step solution

01

Simplify using the Associative Law ONLY

To use the associative law, we can regroup terms within parentheses without changing their order.Start with the inner parentheses: dinner expression is: the inner parentheses: a-b)+b)+b)This simplifies to: a-)+b)+b=b)+bSince it maintains the order of operation.
02

Simplify using the Associative and Distributive Laws ONLY

Applying the associative law:((a-b)+b)+b becomes a-b)+1)+b=1Next,applying the distributive law: Then expression becomes: b)+b=a)(a+)+b
03

Simplify using Associative, Distributive, and Commutative Laws

Finally, employ all three laws:Combine terms. The calculation becomes: b^2= a+b.

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.

Associative Law in Abstract Algebra
The associative law is one of the fundamental rules in abstract algebra. It states that when performing addition or multiplication, the grouping of numbers can be changed without affecting the result. For example, in an addition operation, \( (a + b) + c = a + (b + c)\). Similarly, in multiplication, \( (a \times b) \times c = a \times (b \times c) \). This property allows us to rearrange and simplify complex expressions by focusing on smaller groupings of terms. In our exercise, we applied the associative law to rearrange terms within parentheses to simplify the expression step by step.
Distributive Law in Abstract Algebra
The distributive law bridges addition and multiplication. It states that multiplication distributes over addition or subtraction. For example, \( a \times (b + c) = a \times b + a \times c \). In our given expression \( ((a - b) + b) + b \), we use the distributive law to simplify and combine like terms. The step-by-step solution shows how we can rewrite parts of the expression using the distributive law to make further simplifications by distributing the multiplication over addition or subtraction.
Commutative Law in Abstract Algebra
The commutative law states that the order of numbers in addition or multiplication does not change the result. For addition, this is written as \( a + b = b + a \), and for multiplication, \( a \times b = b \times a \). This law is particularly helpful in rearranging terms for further simplification in algebraic expressions. In our exercise, by using the commutative law along with associative and distributive laws, we finally simplify the expression to its most reduced form. For instance, combining like terms and changing their order to group similar terms together.

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

For any integer, rational, or real number a the following equation holds: \(-a=(-1) \cdot a\). This proposition may seem way too obvious to you, but it's actually saying something very significant. \("-a "\) denotes the additive inverse of \(a\), while " (-1)\(\cdot a "\) denotes the additive inverse of 1 times the number \(a\). There is no a priori reason why theses two things should be the same. Try to think back to when you first learned this arithmetic stuff-at that time, it probably wasn't as obvious as it seems now. The exercise shows that it actually follows from even more basic facts about arithmetic.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.