The question asks to simplify the expression:

$\frac{(a^{-4}b^2c^{-8})^3}{(a^4b^3c^1)^{-2}}$

Let's break it down step by step:

Step 1: Apply the exponents inside the parentheses.

• For the numerator $(a^{-4}b^2c^{-8})^3$: $(a^{-4})^3 = a^{-12}, \quad (b^2)^3 = b^6, \quad (c^{-8})^3 = c^{-24}$ So the numerator becomes: $a^{-12}b^6c^{-24}$

• For the denominator $(a^4b^3c^1)^{-2}$, apply the exponent: $(a^4)^{-2} = a^{-8}, \quad (b^3)^{-2} = b^{-6}, \quad (c^1)^{-2} = c^{-2}$ So the denominator becomes: $a^{-8}b^{-6}c^{-2}$

Step 2: Write the entire expression:

$\frac{a^{-12}b^6c^{-24}}{a^{-8}b^{-6}c^{-2}}$

Step 3: Simplify the expression by subtracting the exponents (since we have division of like bases):

• For $a$: $a^{-12} / a^{-8} = a^{-12 - (-8)} = a^{-4}$
• For $b$: $b^6 / b^{-6} = b^{6 - (-6)} = b^{12}$
• For $c$: $c^{-24} / c^{-2} = c^{-24 - (-2)} = c^{-22}$

Thus, the simplified expression is: $\frac{a^{-4}b^{12}}{c^{22}} = \frac{b^{12}}{a^4c^{22}}$

Answer:

The correct option is C: $\frac{b^{12}}{a^4c^2}$

Would you like to explore any specific part of this solution in more detail?

Here are five related questions for further practice:

1. How do you apply the power rule for exponents in similar expressions?
2. What is the general rule for multiplying exponents with the same base?
3. How would the solution change if one of the exponents was zero?
4. Can you simplify $\frac{(a^3b^{-4})^2}{(a^{-1}b^5)^3}$?
5. How does the order of operations apply when dealing with nested exponents?

Tip: When simplifying expressions with exponents, always handle each variable individually and combine the exponents carefully, remembering to apply the rules of exponent addition or subtraction.