Let's solve the problem step-by-step.
The given problem is to simplify the expression:
[tex]\[
\frac{\frac{a^5 b^3 c}{6 a^2 c^9}}{\frac{2 a^7 b^{11}}{24 c^2}}
\][/tex]
### Step 1: Simplify the First Fraction
First, we simplify the fraction [tex]\(\frac{a^5 b^3 c}{6 a^2 c^9}\)[/tex]:
[tex]\[
\frac{a^5 b^3 c}{6 a^2 c^9} = \frac{a^5}{a^2} \cdot \frac{b^3}{1} \cdot \frac{c}{c^9} \cdot \frac{1}{6} = \frac{a^{5-2}}{1} \cdot \frac{b^3}{1} \cdot \frac{c^{1-9}}{1} \cdot \frac{1}{6} = \frac{a^3 b^3 c^{-8}}{6}
\][/tex]
Hence, the simplified form of the first fraction is:
[tex]\[
\frac{a^3 b^3 c^{-8}}{6}
\][/tex]
### Step 2: Simplify the Second Fraction
Next, we simplify the fraction [tex]\(\frac{2 a^7 b^{11}}{24 c^2}\)[/tex]:
[tex]\[
\frac{2 a^7 b^{11}}{24 c^2} = \frac{2}{24} \cdot a^7 \cdot b^{11} \cdot \frac{1}{c^2}= \frac{1}{12} \cdot a^7 \cdot b^{11} \cdot c^{-2}
\][/tex]
Hence, the simplified form of the second fraction is:
[tex]\[
\frac{a^7 b^{11} c^{-2}}{12}
\][/tex]
### Step 3: Division of the Two Fractions
Next, we divide the first simplified fraction by the second simplified fraction:
[tex]\[
\frac{\frac{a^3 b^3 c^{-8}}{6}}{\frac{a^7 b^{11} c^{-2}}{12}} = \frac{a^3 b^3 c^{-8}}{6} \cdot \frac{12}{a^7 b^{11} c^{-2}}
\][/tex]
### Step 4: Simplifying the Expression
Now we multiply the fractions:
[tex]\[
= \frac{a^3 b^3 c^{-8}}{6} \cdot \frac{12}{a^7 b^{11} c^{-2}} = \frac{a^3 b^3 c^{-8} \cdot 12}{6 a^7 b^{11} c^{-2}}
\][/tex]
Combine and simplify powers of the variables:
[tex]\[
= \frac{12 \cdot a^{3 - 7} \cdot b^{3 - 11} \cdot c^{-8 - (-2)}}{6} = \frac{12 \cdot a^{-4} \cdot b^{-8} \cdot c^{-6}}{6}
\][/tex]
Divide the constants:
[tex]\[
= \frac{12}{6} \cdot a^{-4} \cdot b^{-8} \cdot c^{-6} = 2 \cdot a^{-4} \cdot b^{-8} \cdot c^{-6}
\][/tex]
Express negative exponents as reciprocals:
[tex]\[
= \frac{2}{a^4 b^8 c^6}
\][/tex]
### Final Answer
Hence, the final simplified form of the given expression is:
[tex]\[
\frac{2}{a^4 b^8 c^6}
\][/tex]
