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Chapter 14: Problem 50

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

Short Answer

Expert verified
The expansion and simplification of \((x^{2}+x^{-3})^{4}\) using the Binomial Theorem is \(x^{8} + 4x^{3} + 6x^{-2} + 4x^{-7} + x^{-12}\)

Step by step solution

01

Understand the Binomial Theorem

The Binomial Theorem states that for any number n, \((a+b)^{n} = \sum_{k=0}^{n} {n \choose k}a^{n-k}b^{k}\). Here, \({n \choose k}\) is the binomial coefficient which represents the number of ways to choose k elements from a set of n elements.
02

Apply the Binomial Theorem

Apply the formula to \((x^{2}+x^{-3})^{4}\) and calculate the value of each term. This gives us 5 terms (from k=0 to k=4): \({4 \choose 0}(x^{2})^{4}(x^{-3})^{0} + {4 \choose 1}(x^{2})^{3}(x^{-3})^{1} + {4 \choose 2}(x^{2})^{2}(x^{-3})^{2} + {4 \choose 3}(x^{2})^{1}(x^{-3})^{3} + {4 \choose 4}(x^{2})^{0}(x^{-3})^{4}\)
03

Simplify the expression

Simplify each individual term to obtain the final result. This yields \(x^{8} + 4x^{3} + 6x^{-2} + 4x^{-7} + x^{-12}\)

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Most popular questions from this chapter

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x-2)^{4}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\)

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$5,7,9,11, \dots$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. You are now 25 years old and would like to retire at age 55 with a retirement fund of \(\$ 1,000,000 .\) How much should you deposit at the end of each month for the next 30 years in an IRA paying \(10 \%\) annual interest compounded monthly to achieve your goal? Round to the nearest dollar.

Solve for \(P: A=\frac{P t}{P+t}\)
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