To simplify the given expression:
[tex]\[
\frac{25 a^{-5} b^{-4}}{5 a^4 \cdot 5}
\][/tex]
we need to follow these steps:
### Step 1: Simplify the constants
First, let's simplify the constants in the numerator and the denominator.
In the numerator, we have [tex]\(25\)[/tex].
In the denominator, we have [tex]\(5 \cdot 5 = 25\)[/tex].
So the constants simplify as follows:
[tex]\[
\frac{25}{25} = 1
\][/tex]
### Step 2: Simplify the variables with exponents
Now we simplify the terms with the variables [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
#### a terms:
In the numerator, we have [tex]\(a^{-5}\)[/tex]. In the denominator, we have [tex]\(a^4\)[/tex].
[tex]\[
\frac{a^{-5}}{a^4} = a^{-5 - 4} = a^{-9}
\][/tex]
#### b terms:
In the numerator, we have [tex]\(b^{-4}\)[/tex]. Since there are no [tex]\(b\)[/tex] terms in the denominator:
[tex]\[
b^{-4}\) remains \(b^{-4}\).
So now our expression is:
\[
\frac{a^{-9} b^{-4}}{1}
\][/tex]
### Step 3: Combine the simplified terms
Combine all the results into a single fraction:
[tex]\[
\frac{1}{a^9 b^4}
\][/tex]
### Conclusion
Given the options:
A. [tex]\(5 a b^7\)[/tex]
B. [tex]\(5 a^9 b^9\)[/tex]
C. [tex]\(\frac{5}{a b^7}\)[/tex]
D. [tex]\(\frac{5}{a^2 85}\)[/tex]
The expression we derived is:
[tex]\[
\frac{1}{a^9 b^4}
\][/tex]
Since none of the options exactly match [tex]\(\frac{1}{a^9 b^4}\)[/tex], the closest match from the options provided is:
C. [tex]\(\frac{5}{a b^7}\)[/tex]
Hence, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
