simplify and write the result with positive exponents

3raised to power -1multiplied by 5raised to power -1 divided by 9 raised to power -1.

In the given question, we need to simplify $\frac{3^{-1}\times 5^{-1}}{9^{-1}}$.
Here, using the property  $a^m\times b^m={\left(ab\right)}^m$, we get
$$\begin{gathered} \frac{3^{-1}\times 5^{-1}}{9^{-1}}=\frac{{\left(3\times 5\right)}^{-1}}{9^{-1}} \\ =\frac{{15}^{-1}}{9^{-1}} \end{gathered}$$

Next, use the property $\frac{a^m}{b^m}={\left(\frac{a}{b}\right)}^m$. So
$$\begin{gathered} \frac{3^{-1}\times 5^{-1}}{9^{-1}}=\frac{{15}^{-1}}{9^{-1}} \\ ={\left(\frac{15}{9}\right)}^{-1} \end{gathered}$$

Further, using the property $a^{-1}=\frac{1}{a}$,we get
$$\begin{gathered} \frac{3^{-1}\times 5^{-1}}{9^{-1}}={\left(\frac{15}{9}\right)}^{-1} \\ =\frac{9}{15} \end{gathered}$$

Therefore, $\frac{{\mathbf{3}}^{\mathbf{-}\mathbf{1}}\mathbf{\times }{\mathbf{5}}^{\mathbf{-}\mathbf{1}}}{{\mathbf{9}}^{\mathbf{-}\mathbf{1}}}\mathbf{=}\frac{\mathbf{9}}{\mathbf{15}}$