/*! 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 81 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 81

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There are no values of \(a\) and \(b\) such that $$(a+b)^{4}=a^{4}+b^{4}$$

Short Answer

Expert verified
The statement is generally true, except in the cases when \(a=0\) or \(b=0\)

Step by step solution

01

Expand the left side of the equation

First, you will need to expand \((a+b)^{4}\) using the binomial theorem or otherwise. It can also be calculated as \((a+b)^{2} \cdot (a+b)^{2}\), which equals \(a^{4} + 2a^{3}b + 3a^{2}b^{2} + 2ab^{3} + b^{4}\)
02

Compare the expanded form with the right side of the equation

Now compare the expanded left side of the equation \(a^{4} + 2a^{3}b + 3a^{2}b^{2} + 2ab^{3} + b^{4}\) with the right side of the equation \(a^{4} + b^{4}\). You can see that the two forms are not identical.
03

Evaluate the original statement

Since the left and right sides of the equation aren't equal for generic values of \(a\) and \(b\), it is true that there are no values of \(a\) and \(b\) such that \((a+b)^{4}=a^{4}+b^{4}\). The only exceptions are when \(a=0\) and \(b=0\), in which case the expressions on both sides become zero.
04

Correct the False Statement

The correct statement should be 'There are no values of \(a\) and \(b\), except \(a=0\) and \(b=0\), where \((a+b)^{4}=a^{4}+b^{4}\)'.

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!

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

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely what terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\).

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

What is the common ratio in a geometric sequence?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{4} 3 i+\sum_{i=1}^{4} 4 i=\sum_{i=1}^{4} 7 i$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.