To simplify the given expression:
[tex]\[
\frac{25 a^{-5} b^{-1}}{5 a^4 b}
\][/tex]
we will follow the steps to simplify it step-by-step:
1. Simplify the constants:
[tex]\[
\frac{25}{5} = 5
\][/tex]
2. Simplify the exponents of \(a\):
- In the numerator, the exponent of \(a\) is \(-5\).
- In the denominator, the exponent of \(a\) is \(4\).
- When dividing like bases, we subtract the exponents:
[tex]\[
a^{-5 - 4} = a^{-9}
\][/tex]
3. Simplify the exponents of \(b\):
- In the numerator, the exponent of \(b\) is \(-1\).
- In the denominator, the exponent of \(b\) is \(1\).
- When dividing like bases, we subtract the exponents:
[tex]\[
b^{-1 - 1} = b^{-2}
\][/tex]
4. Combine the simplified results:
Thus, the simplified form of the expression is:
[tex]\[
5 \cdot a^{-9} \cdot b^{-2}
\][/tex]
We can rewrite negative exponents as follows:
[tex]\[
5 \cdot \frac{1}{a^9} \cdot \frac{1}{b^2} = \frac{5}{a^9 b^2}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{5}{a^9 b^2}}
\][/tex]
By referencing the choices, the correct answer corresponds to:
[tex]\[
\boxed{D. \frac{5}{a^2 b^5}}
\][/tex]
