To simplify the expression [tex]\(\left(4 c^4\right)\left(a c^3\right)\left(3 a^5 c\right)\)[/tex], follow these steps:
1. Identify the coefficients:
- The coefficients in the expression are: 4, [tex]\(a\)[/tex] is 1 (since [tex]\(a\)[/tex] does not have a coefficient), and 3.
- Multiply these coefficients together:
[tex]\[
4 \times 1 \times 3 = 12
\][/tex]
2. Multiply the powers of [tex]\(a\)[/tex]:
- In the expression, we have [tex]\(a^1\)[/tex] (or simply [tex]\(a\)[/tex]) and [tex]\(a^5\)[/tex].
- When multiplying like bases, we add the exponents:
[tex]\[
a^1 \times a^5 = a^{1+5} = a^6
\][/tex]
3. Multiply the powers of [tex]\(c\)[/tex]:
- In the expression, we have [tex]\(c^4\)[/tex], [tex]\(c^3\)[/tex], and [tex]\(c^1\)[/tex] (note that [tex]\(c\)[/tex] is the same as [tex]\(c^1\)[/tex]).
- When multiplying like bases, we add the exponents:
[tex]\[
c^4 \times c^3 \times c^1 = c^{4+3+1} = c^8
\][/tex]
Combining all these results together, the simplified expression is:
[tex]\[
12 a^6 c^8
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{12 a^6 c^8}
\][/tex]
