Let's simplify the given mathematical expression step by step:
[tex]\[
\frac{\left(a^2 b^4 c\right)^2\left(6 a^3 b\right)\left(2 c^5\right)^3}{4 a^6 b^{12} c^3}
\][/tex]
### Step 1: Expand each part of the numerator
1. Expanding [tex]\(\left(a^2 b^4 c\right)^2\)[/tex]:
[tex]\[
\left(a^2 b^4 c\right)^2 = a^{2 \cdot 2} b^{4 \cdot 2} c^{1 \cdot 2} = a^4 b^8 c^2
\][/tex]
2. Expanding [tex]\(6 a^3 b\)[/tex]:
[tex]\[
6 a^3 b
\][/tex]
3. Expanding [tex]\(\left(2 c^5\right)^3\)[/tex]:
[tex]\[
\left(2 c^5\right)^3 = 2^3 c^{5 \cdot 3} = 8 c^{15}
\][/tex]
### Step 2: Combine all parts of the numerator
Combine [tex]\(\left(a^2 b^4 c\right)^2\)[/tex], [tex]\(6 a^3 b\)[/tex], and [tex]\(\left(2 c^5\right)^3\)[/tex]:
[tex]\[
a^4 b^8 c^2 \cdot 6 a^3 b \cdot 8 c^{15} = 48 a^7 b^9 c^{17}
\][/tex]
### Step 3: Expand the denominator
1. Expanding [tex]\(4 a^6 b^{12} c^3\)[/tex]:
[tex]\[
4 a^6 b^{12} c^3
\][/tex]
### Step 4: Form the new fraction and simplify
Combine the expanded numerator and denominator:
[tex]\[
\frac{48 a^7 b^9 c^{17}}{4 a^6 b^{12} c^3}
\][/tex]
Divide the coefficients and subtract the exponents of the variables:
[tex]\[
\frac{48}{4} \cdot \frac{a^7}{a^6} \cdot \frac{b^9}{b^{12}} \cdot \frac{c^{17}}{c^3} = 12 \cdot a \cdot b^{-3} \cdot c^{14}
\][/tex]
Rewrite using positive exponents:
[tex]\[
12 \cdot a \cdot \frac{c^{14}}{b^3} = \frac{12 a c^{14}}{b^3}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\boxed{\frac{12 a c^{14}}{b^3}}
\][/tex]
