The problem is asking to simplify the expression:

$\left( \frac{12p^{-6}}{c^{-3}x^4} \right) \times \left( \frac{2p^8}{c^{-4}x^6} \right)$

Step 1: Simplify the coefficients

Multiplying the numerical coefficients: $12 \times 2 = 24$

Step 2: Simplify the powers of $p$

Using the law of exponents $p^a \times p^b = p^{a+b}$: $p^{-6} \times p^8 = p^{-6 + 8} = p^2$

Step 3: Simplify the powers of $c$

Using the same law for $c$: $c^{-3} \times c^{-4} = c^{-3 + (-4)} = c^{-7}$

Step 4: Simplify the powers of $x$

$x^4 \times x^6 = x^{4+6} = x^{10}$

Final Simplified Expression:

$\frac{24p^2}{c^{-7}x^{10}} = 24p^2c^7x^{-10}$

Thus, the simplified expression is: $24p^2c^7x^{-10}$

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:

1. How do you handle negative exponents in algebraic expressions?
2. Can you explain the multiplication of variables with different bases?
3. What are the laws of exponents, and how do they apply to division?
4. How do you simplify an expression with fractions and exponents?
5. What is the general rule for multiplying and dividing expressions with exponents?

Tip: Always group terms with the same base when simplifying expressions to avoid errors in combining exponents!