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Chapter 14: Problem 49

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{3}+x^{-2}\right)^{4}$$

Short Answer

Expert verified
\(x^{12} + 4x^{5} + 6x^{-2} + 4x^{-8} + x^{-8}\)

Step by step solution

01

Apply the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k} b^{k}\). Here, our \(a\) is \(x^{3}\) and \(b\) is \(x^{-2}\), and \(n\) is 4. We will use this formula to expand the given expression.
02

Calculate the terms

Evaluate the Binomial coefficients (n choose k) for \(k = 0\) to \(4\) and substitute for \(a\) and \(b\) to get the terms of our expanded expression. The terms are: \[{4\choose 0} (x^{3})^{4} (x^{-2})^{0}\] \[{4\choose 1} (x^{3})^{3} (x^{-2})^{1}\] \[{4\choose 2} (x^{3})^{2} (x^{-2})^{2}\] \[{4\choose 3} (x^{3})^{1} (x^{-2})^{3}\] \[{4\choose 4} (x^{3})^{0} (x^{-2})^{4}\].
03

Simplify the terms

Evaluate coefficients and simplify each term by multiplying exponents: \(1.x^{12}\) \(4x^{5}\) \(6x^{-2} \) \(4x^{-8} \) \(1x^{-8}\)
04

Write the final expanded expression

Add all the terms to write our final expanded expression: \(x^{12} + 4x^{5} + 6x^{-2} + 4x^{-8} + x^{-8}\).

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Most popular questions from this chapter

A deposit of 10,000 dollars is made in an account that earns \(8 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=10,000\left(1+\frac{0.08}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after six years. Round to the nearest cent.

What is the meaning of the symbol \Sigma? Give an example with your description.

What is the difference between a geometric sequence and an infinite geometric series?

Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. \begin{aligned}&16, \quad\quad\quad\quad\quad 0.96(16), \quad(0.96)^{2}(16), \quad(0.96)^{3}(16), \ldots\\\&\begin{array}{|l|l|l|}\hline \begin{array}{l}\text { 1st swing } \\\\\end{array} & \begin{array}{l}\text{ 2nd swing } \\\\\end{array} & \begin{array}{l}\text { 3rd swing } \\\\\end{array} & \begin{array}{l}\text { 4th swing } \\\\\end{array} \\\\\hline\end{array}\end{aligned} After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$\frac{4}{1}, \frac{9}{2}, \frac{16}{3}, \frac{25}{4}, \dots$$
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