5.14. \( \left(\frac{3 x^{-6}}{9 x^{-3}}\right)^{-2} \quad \) 5.15. \( \frac{x+y}{x^{-1}+y^{-1}} \)

Let's start by simplifying the first expression, \( \left(\frac{3 x^{-6}}{9 x^{-3}}\right)^{-2} \). We can simplify the fraction inside the parentheses first. This gives us \( \frac{3}{9} \cdot \frac{x^{-6}}{x^{-3}} = \frac{1}{3} \cdot x^{-6 + 3} = \frac{1}{3} \cdot x^{-3} \). Now we raise this to the power of -2: \[ \left(\frac{1}{3} \cdot x^{-3}\right)^{-2} = \left(\frac{1}{3}\right)^{-2} \cdot (x^{-3})^{-2} = 9 \cdot x^{6} = 9x^6. \] Next, for the second expression \( \frac{x+y}{x^{-1}+y^{-1}} \), we start by rewriting the denominators: \[ x^{-1} + y^{-1} = \frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}. \] So, our expression transforms to: \[ \frac{x+y}{\frac{y+x}{xy}} = \frac{(x+y) \cdot xy}{y+x} = xy. \] Therefore, the simplified forms are \( 9x^6 \) for the first expression and \( xy \) for the second.