Let's simplify the given expression:

\[
\frac{8c^3 \times 12c^6}{6c^7 \times 3c^4}
\]

### Step 1: Simplify the coefficients
\[
\frac{8 \times 12}{6 \times 3} = \frac{96}{18} = \frac{16}{3}
\]

### Step 2: Simplify the powers of \( c \)
For the numerator: \( c^3 \times c^6 = c^{3+6} = c^9 \)

For the denominator: \( c^7 \times c^4 = c^{7+4} = c^{11} \)

### Step 3: Simplify the powers of \( c \) in the fraction
\[
\frac{c^9}{c^{11}} = c^{9-11} = c^{-2}
\]

### Step 4: Combine the simplified coefficient and the power of \( c \)
\[
\frac{16}{3} \times c^{-2} = \frac{16}{3c^2}
\]

### Final Answer:
\[
\frac{16}{3c^2}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you simplify expressions involving negative exponents?
2. What happens if the powers of \(c\) in the numerator are larger than in the denominator?
3. Can you apply the same rules of exponents to other variables?
4. How do you simplify a more complex fraction with multiple terms?
5. How would the result change if you had variables like \( x \) in the expression as well?

**Tip:** Always simplify coefficients and variables separately before combining the results.

Learn how to simplify the rational expression 8c^3 * 12c^6 / 6c^7 * 3c^4 step by step. This algebra problem involves simplifying coefficients and applying exponent rules. Ideal for students in Grades 8-10, this guide covers the laws of exponents and simplifying algebraic expressions.

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