Expand the expression \((\frac{x}{3} + \frac{1}{x})^5\).
Solution and Explanation
By using Binomial Theorem, the expression \((\frac{x}{3} + \frac{1}{x})^5\) can be expanded as
\((\frac{x}{3} + \frac{1}{x})^5\) = \(^5C_0(\frac{x}{3})^5 + ^5C_1(\frac{x}{3})^4(\frac{1}{x}) + ^5C_2 (\frac{x}{3})^3(\frac{1}{x})^2 + ^5C_3(\frac{x}{3})^2(\frac{1}{x})^3 + ^5C_4(\frac{x}{3})(\frac{1}{x})^4+ ^5C_5(\frac{1}{x})^5\)
= \(\frac{x^5}{ 243} + 5(\frac{x^4}{81})(\frac{1}{x}) + 10 (\frac{x^3}{27})(\frac{1}{x^2}) + 10 (\frac{x^2}{9})(\frac{1}{x^3}) + 5 (\frac{x}{3})(\frac{1}{x^4}) +(\frac{1}{x^5})\)
= \(\frac{x^5}{ 243} + \frac{5x^3}{81} + \frac{10x}{ 27} + \frac{10}{9x} + \frac{5}{ 3x^3} + \frac{1}{x^5}\).
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