To evaluate the expression \( \frac{y^5 - 12}{y(z^2 - 5)} \) when \( y = 2 \) and \( z = 5 \), we can substitute the values of \( y \) and \( z \) into the expression.
- Substitute \( y = 2 \) and \( z = 5 \):
\[ \text{Expression} = \frac{2^5 - 12}{2(5^2 - 5)} \]
- Calculate \( 2^5 \):
\[ 2^5 = 32 \]
- Now, substitute this back into the expression:
\[ = \frac{32 - 12}{2(5^2 - 5)} \]
- Simplify \( 32 - 12 \):
\[ = \frac{20}{2(5^2 - 5)} \]
- Now calculate \( 5^2 - 5 \):
\[ 5^2 = 25 \quad \Rightarrow \quad 25 - 5 = 20 \]
- Substitute this back into the expression:
\[ = \frac{20}{2(20)} \]
- Calculate \( 2(20) \):
\[ = 40 \]
- Now, substitute this into the expression:
\[ = \frac{20}{40} \]
- Simplify \( \frac{20}{40} \):
\[ = \frac{1}{2} \]
Therefore, the final result is:
\[ \frac{1}{2} \]