Let's simplify the given expression step by step. The expression is:

$\frac{(-2k^{-3})^2 (6k^2)^4}{(9k^4)^2}$

Step 1: Simplify each part individually

• For $(-2k^{-3})^2$: $(-2)^2 = 4 \quad \text{and} \quad (k^{-3})^2 = k^{-6}$ So, $(-2k^{-3})^2 = 4k^{-6}$.

• For $(6k^2)^4$: $6^4 = 1296 \quad \text{and} \quad (k^2)^4 = k^8$ So, $(6k^2)^4 = 1296k^8$.

• For $(9k^4)^2$: $9^2 = 81 \quad \text{and} \quad (k^4)^2 = k^8$ So, $(9k^4)^2 = 81k^8$.

Step 2: Substitute back into the original expression

The expression becomes: $\frac{4k^{-6} \cdot 1296k^8}{81k^8}$

Step 3: Simplify the constants

$\frac{4 \cdot 1296}{81} = \frac{5184}{81} = 64$

Step 4: Simplify the exponents of $k$

• In the numerator: $k^{-6} \cdot k^8 = k^{(-6 + 8)} = k^2$
• In the denominator: $k^8$

So, the $k$-terms become: $\frac{k^2}{k^8} = k^{2-8} = k^{-6}$

Step 5: Final result

The simplified expression is: $64k^{-6} = \frac{64}{k^6}$

Thus, the simplified form of the given expression is:

$\boxed{\frac{64}{k^6}}$

Would you like further details or have any questions?

Related Questions:

1. How do you simplify an expression using the properties of exponents?
2. What is the process of simplifying negative exponents?
3. Can you explain how to multiply powers with the same base?
4. How do you approach simplifying expressions with both negative and positive exponents?
5. How does simplifying a fraction with powers work in general?

Tip:

Always apply the power rule when simplifying terms with exponents, remembering that $(a^m)^n = a^{m \cdot n}$.