To find the value of the question mark in the rational expression \( \frac{19}{z^2 - 7z} = \frac{?}{z(z - 7)(z - 5)} \), we first need to simplify \( z^2 - 7z \).
We can factor \( z^2 - 7z \) as follows:
\[ z^2 - 7z = z(z - 7) \]
Now we set up the equation:
\[ \frac{19}{z(z - 7)} = \frac{?}{z(z - 7)(z - 5)} \]
To find the numerator represented by the question mark, we can cross multiply:
\[ 19 \cdot (z(z - 7)(z - 5)) = ? \cdot (z(z - 7)) \]
This means we need to express \( 19(z(z - 7)(z - 5)) \) divided by \( z(z - 7) \):
\[ ? = \frac{19 \cdot z(z - 7)(z - 5)}{z(z - 7)} \]
Since \( z(z - 7) \) in the numerator and the denominator cancel out, we have:
\[ ? = 19(z - 5) \]
Thus, the final answer is:
\[ ? = 19z - 95 \]
So, the rational expression with the indicated denominator is:
\[ \frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{19z - 95}{z(z - 7)(z - 5)} \]