$\displaystyle z=\frac{3+4i}{2-3i}$. What is the complex number which satisfies the equation $\displaystyle zw=1$.

My attempt:

$\displaystyle \begin{array}{rcrcrc}

z=\frac{3+4i}{2-3i}\\

\\

zw=1\\

\\

w=\frac{1}{z}\\

\\

\frac{3+4i}{2-3i}*\frac{2+3i}{2+3i}=\frac{-6+17i}{13}

\end{array}$

Is it right up to there? If so, I see I need to get the reciprocal of that (which will be the value of w). How would I do that?