Set the function equal to zero and clear the denominator by multiplying by the least common denominator, \(\left( {z - 5} \right)\left( {z - 8} \right)\), and then solve the resulting equation.

\[\begin{align*}\left( {z - 5} \right)\left( {z - 8} \right)\left( {\frac{z}{{z - 5}} - \frac{4}{{z - 8}}} \right) & = 0\\ z\left( {z - 8} \right) - 4\left( {z - 5} \right) & = 0\\ {z^2} - 12z + 20 & = 0\\ \left( {z - 10} \right)\left( {z - 2} \right) & = 0\end{align*}\]

So, it looks like the function has two roots, \(z = 2\) and \(z = 10\) however recall that because we started off with a function that contained rational expressions we need to go back to the original function and make sure that neither of these will create a division by zero problem in the original function. In this case neither do and so the two roots are,

\[z = 2\hspace{0.25in}\hspace{0.25in}z = 10\]