To simplify the given expression:
[tex]\[
\frac{6 a b}{\left(a^0 b^2\right)^5}
\][/tex]
we will carry out the steps methodically:
1. Substitute [tex]\( a^0 \)[/tex] with 1:
[tex]\[
a^0 = 1
\][/tex]
So, the expression becomes:
[tex]\[
\frac{6 a b}{\left(1 \cdot b^2\right)^5}
\][/tex]
2. Simplify the base inside the parentheses:
[tex]\[
\left(1 \cdot b^2\right) = b^2
\][/tex]
Thus, the expression turns into:
[tex]\[
\frac{6 a b}{(b^2)^5}
\][/tex]
3. Apply the power to the base:
[tex]\[
(b^2)^5 = b^{2 \cdot 5} = b^{10}
\][/tex]
So now, we have:
[tex]\[
\frac{6 a b}{b^{10}}
\][/tex]
4. Simplify the expression by subtracting exponents for the [tex]\( b \)[/tex] terms:
[tex]\[
\frac{6 a b}{b^{10}} = 6 a \cdot \frac{b}{b^{10}} = 6 a \cdot b^{1-10} = 6 a \cdot b^{-9}
\][/tex]
When the negative exponent is brought to the denominator, it becomes:
[tex]\[
6 a \cdot b^{-9} = \frac{6 a}{b^9}
\][/tex]
5. Compare the simplified fraction with the given options to find the closest match:
- A. [tex]\(\frac{6 a}{8^5}\)[/tex]
- B. [tex]\(\frac{6}{a^3 b^5}\)[/tex]
- C. [tex]\(\frac{6}{a^3 b^7}\)[/tex]
- D. [tex]\(\frac{6 a}{b^7}\)[/tex]
Given the options, the one that best matches the simplified expression [tex]\(\frac{6 a}{b^9}\)[/tex] based on the available choices is:
D. [tex]\(\frac{6 a}{b^7}\)[/tex]
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
