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Chapter 14: Problem 51

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^{3}$$

Short Answer

Expert verified
The simplified form of \(\left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^3\) is \(x-x\sqrt[3]{x^{-2}}+3\sqrt[3]{x^{-1}}-1\).

Step by step solution

01

Identify the Terms

In this problem, \(a=x^{\frac{1}{3}}\), \(b=-x^{-\frac{1}{3}}\) and \(n=3\). So the expression to expand is \(\left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^{3}\).
02

Use the Binomial Theorem

Use the expansion \((a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}\) but with \(a=x^{\frac{1}{3}}\) and \(b=-x^{-\frac{1}{3}}\) and give each term for \(k=0\) to \(k=3\). So, the expanded form becomes \[\binom{3}{0}(x^{\frac{1}{3}})^{3-0}(-x^{-\frac{1}{3}})^{0}+\binom{3}{1}(x^{\frac{1}{3}})^{3-1}(-x^{-\frac{1}{3}})^{1}+\binom{3}{2}(x^{\frac{1}{3}})^{3-2}(-x^{-\frac{1}{3}})^{2}+\binom{3}{3}(x^{\frac{1}{3}})^{3-3}(-x^{-\frac{1}{3}})^{3}\]
03

Simplify the Expression

Expand the binomial coefficients and simplify each term. This yields the simplified form \[x-x\sqrt[3]{x^{-2}}+3\sqrt[3]{x^{-1}}-1\].

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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. $$f_{1}(x)=(x-2)^{4}$$

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