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Let's solve the given expression:

\[
\frac{\left(a^2 b^4 c\right)^2 \left(6a^3 b\right) \left(2c^5\right)^3}{4a^6 b^{12} c^3}
\]

### Step-by-step simplification:

#### 1. Simplifying each term:

- \(\left(a^2 b^4 c\right)^2\):
  \[
  (a^2 b^4 c)^2 = a^{2 \times 2} b^{4 \times 2} c^{1 \times 2} = a^4 b^8 c^2
  \]

- \(\left(6a^3 b\right)\) is already in its simplest form:
  \[
  6a^3 b
  \]

- \(\left(2c^5\right)^3\):
  \[
  (2c^5)^3 = 2^3 c^{5 \times 3} = 8c^{15}
  \]

#### 2. Multiply the numerators:

\[
(a^4 b^8 c^2)(6a^3 b)(8c^{15}) = 6 \times 8 \times a^{4+3} \times b^{8+1} \times c^{2+15}
\]

\[
= 48a^7 b^9 c^{17}
\]

#### 3. Now divide by the denominator:

Denominator: \(4a^6 b^{12} c^3\)

Now, divide each part:

\[
\frac{48a^7 b^9 c^{17}}{4a^6 b^{12} c^3} = \frac{48}{4} \times \frac{a^7}{a^6} \times \frac{b^9}{b^{12}} \times \frac{c^{17}}{c^3}
\]

\[
= 12 \times a^{7-6} \times b^{9-12} \times c^{17-3}
\]

\[
= 12a^1 b^{-3} c^{14}
\]

\[
= \frac{12a c^{14}}{b^3}
\]

#### 4. Final Answer:

The equivalent expression is:

\[
\boxed{\frac{12ac^{14}}{b^3}}
\]

Thus, the correct answer is the third option:
\[
\frac{12ac^{14}}{b^3}
\]

---

Would you like any further details or have any questions?

Here are 5 related questions to expand this concept:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for multiplying powers with the same base?
3. How do you simplify complex fractions with multiple variables?
4. What is the best strategy for simplifying exponential expressions inside parentheses?
5. Can you apply these rules to expressions with radicals?

**Tip:** When simplifying exponents, always perform operations with the same base first before combining terms from other bases.

Which expression is equivalent to \frac{\left(a^2 b^4 c\right)^2 \left(6a^3 b\right) \left(2c^5\right)^3}{4a^6 b^{12} c^3}?

Learn how to simplify the expression \frac{\left(a^2 b^4 c\right)^2 \left(6a^3 b\right) \left(2c^5\right)^3}{4a^6 b^{12} c^3} step-by-step using exponent rules. This problem covers key algebraic concepts like multiplication, division, and power of exponents. Suitable for students in Grades 10-12.

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