Answer
$\displaystyle ( \frac{1}{x}-\sqrt{x})^{5}=\frac{1}{x^{5}}-\frac{5}{x^{7/2}}+\frac{10}{x^{2}}-\frac{10}{x^{1/2}}+5x-x^{5/2}$
Or with negative exponents:
$\displaystyle ( \frac{1}{x}-\sqrt{x})^{5}=x^{-5}-5x^{-7/2}+10x^{-2}-10x^{-1/2}+5x-x^{5/2}$
Work Step by Step
We use the values from the 5th row of Pascal's Triangle to find the coefficients:
$\displaystyle ( \frac{1}{x}-\sqrt{x})^{5}=(\frac{1}{x})^{5}-5(\frac{1}{x})^{4}\sqrt{x}+10(\frac{1}{x})^{3}x-10(\frac{1}{x})^{2}x\sqrt{x}+5(\frac{1}{x})^1x^{2}-x^{2}\sqrt{x}
=\frac{1}{x^{5}}-\frac{5}{x^{7/2}}+\frac{10}{x^{2}}-\frac{10}{x^{1/2}}+5x-x^{5/2}$
Or with negative exponents:
$\displaystyle=x^{-5}-5x^{-7/2}+10x^{-2}-10x^{-1/2}+5x-x^{5/2}$
