We have to solve:
[tex]\begin{gathered} \frac{2}{3z}-(\frac{5}{6z^2}-\frac{z+3}{z}) \\ \frac{2}{3z}-(\frac{5}{6z^2}-\frac{6z}{6z}\cdot\frac{z+3}{z}) \\ \frac{2}{3z}-(\frac{5}{6z^2}-\frac{6z(z+3)}{6z^2}) \\ \frac{2}{3z}-(\frac{5-6z(z+3)}{6z^2}) \\ \frac{2}{3z}-(\frac{5-6z^2-18z}{6z^2}) \\ \frac{2}{3z}\cdot\frac{2z}{2z}-(\frac{5-6z^2-18z}{6z^2}) \\ \frac{4z}{6z^2}-(\frac{5-6z^2-18z}{6z^2}) \\ \frac{4z-5+6z^2+18z}{6z^2} \\ \frac{6z^2+22z-5}{6z^2} \end{gathered}[/tex]
For the first factor, when we have to substract the fractions inside the parenthesis, we have to find the common factors between 6z^2 and z. In this case, z is part of 6z^2, so we have to multiply z by 6z^2/z=6z.
If we multiply the fraction by 6z/6z we don't change the result and get the same denominator in order to do teh substraction.
In the 6th step, we have to convert the denominator 3z into 6z^2. We can find the factor by dividing 6z^2 by 3z:
[tex]\frac{6z^2}{3z}=\frac{6}{3}\cdot\frac{z^2}{z}=2z[/tex]
With this factor, 2z, we can convert the denominator from 3z to 6z^2 and perform the operation.
Answer: (6z^2+22z-5)/6z^2
