((230a^15bc^3)/(17ab^4c^2)) / ((67abd^4)/(13a^2b^2c^3))

Let's simplify the given expression:

\[
\frac{\left(\frac{230a^{15}bc^3}{17ab^4c^2}\right)}{\left(\frac{67abd^4}{13a^2b^2c^3}\right)}
\]

We can simplify the expression step by step.

### Step 1: Simplify the numerator and the denominator separately.
- **Numerator:**
\[
\frac{230a^{15}bc^3}{17ab^4c^2}
\]
Simplify by canceling out common factors:
\[
= \frac{230}{17} \cdot \frac{a^{15}}{a^1} \cdot \frac{b^1}{b^4} \cdot \frac{c^3}{c^2}
\]
\[
= \frac{230}{17} \cdot a^{14} \cdot b^{-3} \cdot c^1
\]
\[
= \frac{230a^{14}c}{17b^3}
\]

- **Denominator:**
\[
\frac{67abd^4}{13a^2b^2c^3}
\]
Simplify by canceling out common factors:
\[
= \frac{67}{13} \cdot \frac{a^1}{a^2} \cdot \frac{b^1}{b^2} \cdot \frac{d^4}{c^3}
\]
\[
= \frac{67}{13} \cdot a^{-1} \cdot b^{-1} \cdot \frac{d^4}{c^3}
\]
\[
= \frac{67d^4}{13ab \cdot c^3}
\]

### Step 2: Divide the simplified expressions.
Now, divide the simplified numerator by the simplified denominator:

\[
\frac{\frac{230a^{14}c}{17b^3}}{\frac{67d^4}{13ab \cdot c^3}}
\]

This is equivalent to multiplying by the reciprocal of the denominator:

\[
= \frac{230a^{14}c}{17b^3} \times \frac{13ab \cdot c^3}{67d^4}
\]

### Step 3: Simplify the resulting expression.
Multiply the numerators and denominators:

\[
= \frac{230 \times 13 \times a^{14} \times a \times c \times c^3}{17 \times 67 \times b^3 \times b \times d^4}
\]
\[
= \frac{2990a^{15}c^4}{1139b^4d^4}
\]

### Final Answer:
\[
\frac{2990a^{15}c^4}{1139b^4d^4}
\]

Would you like any further details on this problem or have any additional questions?

### Related Questions:
1. How would the expression change if the variables \(a\) and \(c\) had negative exponents?
2. What would happen if the term \(d^4\) was not present in the denominator?
3. Can you simplify the expression if additional terms were added to both the numerator and denominator?
4. How would you handle this problem if all variables had different coefficients?
5. Can you factor any terms from the final simplified expression?

**Tip:** When dividing fractions, always remember that dividing by a fraction is the same as multiplying by its reciprocal.

Learn how to simplify the complex rational expression ((230a^15bc^3)/(17ab^4c^2)) / ((67abd^4)/(13a^2b^2c^3)) step by step. This problem covers key algebraic concepts, including the division of fractions, properties of exponents, and the simplification of rational expressions. Suitable for students in Grades 9-11.

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