Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Binomial Expansion Challenge**
**Problem Statement:**
Find the indicated term of the binomial expansion.
\[
\left( \sqrt{2} u^2 + v^3 \right)^9
\]
**Objective:**
Identify the second term of this expansion.
---
**Solution Guide:**
To solve this, apply the binomial expansion formula:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For this specific problem, set:
- \( a = \sqrt{2} u^2 \)
- \( b = v^3 \)
- \( n = 9 \)
The general term in the expansion is given by:
\[
\binom{9}{k} (\sqrt{2} u^2)^{9-k} (v^3)^k
\]
To determine the second term, use \( k = 1 \):
\[
\binom{9}{1} (\sqrt{2} u^2)^{8} (v^3)^1
\]
Simplify:
\[
\binom{9}{1} \cdot (\sqrt{2})^{8} \cdot (u^2)^8 \cdot v^3
\]
\[
= 9 (\sqrt{2})^8 u^{16} v^3
\]
Calculate \((\sqrt{2})^8\):
\[
(\sqrt{2})^8 = (2^4) = 16
\]
Thus, the second term is:
\[
9 \cdot 16 \cdot u^{16} \cdot v^3 = 144 u^{16} v^3
\]
This is the second term of the binomial expansion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe20bf933-237a-4bac-9ccb-ad53d7fa80c9%2F7d7072c0-8a4c-4eb5-9e1b-f11477e6c768%2Figh1ep_processed.png&w=3840&q=75)
Transcribed Image Text:**Binomial Expansion Challenge**
**Problem Statement:**
Find the indicated term of the binomial expansion.
\[
\left( \sqrt{2} u^2 + v^3 \right)^9
\]
**Objective:**
Identify the second term of this expansion.
---
**Solution Guide:**
To solve this, apply the binomial expansion formula:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For this specific problem, set:
- \( a = \sqrt{2} u^2 \)
- \( b = v^3 \)
- \( n = 9 \)
The general term in the expansion is given by:
\[
\binom{9}{k} (\sqrt{2} u^2)^{9-k} (v^3)^k
\]
To determine the second term, use \( k = 1 \):
\[
\binom{9}{1} (\sqrt{2} u^2)^{8} (v^3)^1
\]
Simplify:
\[
\binom{9}{1} \cdot (\sqrt{2})^{8} \cdot (u^2)^8 \cdot v^3
\]
\[
= 9 (\sqrt{2})^8 u^{16} v^3
\]
Calculate \((\sqrt{2})^8\):
\[
(\sqrt{2})^8 = (2^4) = 16
\]
Thus, the second term is:
\[
9 \cdot 16 \cdot u^{16} \cdot v^3 = 144 u^{16} v^3
\]
This is the second term of the binomial expansion.
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