Let’s simplify the given expression step by step.

The problem is to simplify:

$\frac{(a^3b^4c^2)^2}{ab^4c^3}$

Step 1: Expand the numerator

Using the power rule $(x^m)^n = x^{m \cdot n}$, expand $(a^3b^4c^2)^2$:

$(a^3b^4c^2)^2 = a^{3 \cdot 2}b^{4 \cdot 2}c^{2 \cdot 2} = a^6b^8c^4$

So the numerator becomes $a^6b^8c^4$.

Step 2: Simplify the division

Divide the expanded numerator by the denominator $ab^4c^3$:

$\frac{a^6b^8c^4}{ab^4c^3}$

Using the division rule $x^m / x^n = x^{m-n}$, simplify each variable:

1. For $a$: $a^6 / a = a^{6-1} = a^5$
2. For $b$: $b^8 / b^4 = b^{8-4} = b^4$
3. For $c$: $c^4 / c^3 = c^{4-3} = c^1 = c$

Final Answer:

The simplified expression is:

$a^5b^4c$

The correct option is:

A. $a^5b^4c$

Would you like further details or assistance with related concepts? Here are some questions for deeper exploration:

1. What are the general rules of exponents used in simplification?
2. How does the power rule apply to multiple variables in a term?
3. What is the difference between simplifying powers and multiplying powers?
4. Can you explain why $c^{4-3} = c^1$ and not just $c$?
5. How do we handle cases where the exponent becomes negative in division?

Tip: Always apply exponent rules systematically to avoid missing terms.