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Chapter 11: Problem 354

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The eighth term of \((7+5 y)^{14}\)

Short Answer

Expert verified
The eighth term is \(282475249000000y^7\).

Step by step solution

01

Identify the General Term Formula for a Binomial Expansion

In a binomial expansion of the form \((a + b)^n\), the general term, often called the \(k\)-th term, is given by: \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). In this problem, \(a = 7\), \(b = 5y\), and \(n = 14\).
02

Determine the Desired Term Number

We are asked to find the eighth term of the expansion. In the general term formula, the term index \(k\) is calculated based on \(k+1\). Thus, for the eighth term, \(k = 8 - 1 = 7\).
03

Calculate the Numerical Coefficient with Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) for \(n = 14\) and \(k = 7\) is calculated as:\[ \binom{14}{7} = \frac{14!}{7!(14-7)!} = 3432. \]
04

Calculate the Powers of the Terms in the Expansion

For the term \((7 + 5y)^{14}\), evaluate the powers: \(a^{n-k} = 7^{14-7} = 7^7\) and \(b^k = (5y)^7\).
05

Evaluate the Powers and Calculate the Complete Term

First, find \(7^7\):\[7^7 = 823543.\]Next, evaluate \((5y)^7\):\[(5y)^7 = 5^7 \cdot y^7 = 78125y^7.\]Multiply these results by the binomial coefficient (3432) to form the term:\[3432 \cdot 823543 \cdot 78125y^7.\]
06

Calculate the Full Result for the Eighth Term

The eighth term is:\[T_8 = 3432 \cdot 823543 \cdot 78125y^7.\]Calculate the product to get the coefficient of \(y^7\). Finally, multiply all numeric values:\[282475249000000y^7.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficient
The binomial coefficient, often denoted as \( \binom{n}{k} \), is a crucial component in binomial expansion. It represents the number of ways to choose \( k \) elements from a set of \( n \) elements and is calculated using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, \(!\) denotes factorial, which means multiplying a sequence of descending natural numbers. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
This coefficient indicates the weight applied to each term in the binomial expansion.
  • In the given exercise, \( n = 14 \) and \( k = 7 \) for the eighth term, resulting in \( \binom{14}{7} = 3432 \).
  • This value is essential for calculating the entire term in the expansion.
General Term Formula
To efficiently find a specific term in a binomial expansion, rather than expanding everything, we use the general term formula:
\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]This formula helps determine any term directly by plugging in the respective values.
For any binomial \((a + b)^n\), you just need:
  • \( \binom{n}{k} \) - the binomial coefficient.
  • \( a^{n-k} \) - the power of the first term.
  • \( b^k \) - the power of the second term.
In the exercise, where \( (7+5y)^{14} \), to find the eighth term, we use \( k = 7 \) as follows:
  • First calculate \( \binom{14}{7} = 3432 \).
  • Then, find \( 7^{14-7} = 7^7 \).
  • Finally, calculate \( (5y)^7 = 5^7 \cdot y^7 \).
By multiplying these together, you obtain the term value without full expansion.
Exponentiation
Exponentiation involves raising a number to a power, which is a key operation in binomial expansions. It鈥檚 expressed as \( a^b \), where \( a \) is the base and \( b \) is the exponent.
This operation indicates how many times to multiply the base by itself.
In our problem example:
  • We calculate \( 7^7 \), which involves multiplying 7 by itself seven times, resulting in 823,543.
  • For \((5y)^7\), the base \( 5y \) is multiplied by itself seven times: \(5^7 \cdot y^7 \).
  • Calculating \(5^7\) results in 78,125, so \((5y)^7 = 78,125y^7\).
These exponentiations are then multiplied with the binomial coefficient to derive specific terms within the expansion. Exponentiation thus provides the structure of each term鈥檚 power distribution in the binomial expansion, making it a fundamental operation in solving such problems efficiently.

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Most popular questions from this chapter

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