To write the rational expression \( \frac{19}{z^2 - 7z} \) with the denominator \( z(z - 7)(z - 5) \), we first need to factor the original denominator.
The expression \( z^2 - 7z \) can be factored as follows:
\[ z^2 - 7z = z(z - 7) \]
Now, we need to convert \( \frac{19}{z(z - 7)} \) into a form where the denominator is \( z(z - 7)(z - 5) \). We can do this by multiplying both the numerator and the denominator by \( (z - 5) \):
\[ \frac{19}{z(z - 7)} \cdot \frac{(z - 5)}{(z - 5)} = \frac{19(z - 5)}{z(z - 7)(z - 5)} \]
Now simplifying the numerator:
\[ 19(z - 5) = 19z - 95 \]
Putting it all together, we have:
\[ \frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{19z - 95}{z(z - 7)(z - 5)} \]
Thus, the rational expression with the indicated denominator is:
\[ \frac{19z - 95}{z(z - 7)(z - 5)} \]