/*! 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 29 Use the Binomial Theorem to expa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 29

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (2 a+b)^{6} $$

Short Answer

Expert verified
The expanded form of \((2a + b)^6\) is \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\).

Step by step solution

01

Understanding the Binomial Theorem

The Binomial Theorem tells us how to expand a binomial raised to a certain power. For any natural number \( n \), and any real numbers \( a \) and \( b \), \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k}b^{k}\).
02

Application of the Binomial Theorem

Now we put \( a \) as \( 2a \), \( b \) as \( b \), and \( n \) as \( 6 \) into the formula given by the Binomial Theorem. Following the formula, we find the expansion of \((2a + b)^6\) to be \(\sum_{k=0}^{6} {6\choose k} (2a)^{6-k} b^{k}\).
03

Calculation and Simplification

Applying the equation from step 2, it becomes: \({6\choose0} (2a)^6 b^0 + {6\choose1} (2a)^5 b^1 + {6\choose2} (2a)^4 b^2 + {6\choose3} (2a)^3 b^3 + {6\choose4} (2a)^2 b^4 + {6\choose5} (2a)^1 b^5 + {6\choose6} (2a)^0 b^6\). This simplifies to \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\).

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

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a picture card.}$$

Explaining the Concepts Describe the difference between theoretical probability and empirical probability.

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a. a Democrat who is not a business major. b. a student who is neither a Democrat nor a business major.

Use the formula for the value of an annuity to solve Exercises 77鈥�84. Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 75\) at the end of each month in an IRA that pays \(6.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) b. Find the interest.

Graph \(y=3 \tan \frac{x}{2}\) for \(-\pi
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.