We are asked to simplify the expression $(\frac{-9k^{10}}{-3k^4})^3$.

1. Problem Description

We are asked to simplify the expression $(\frac{-9k^{10}}{-3k^4})^3$.

2. Solution Steps

First, simplify the fraction inside the parentheses:

$\frac{-9k^{10}}{-3k^4} = \frac{-9}{-3} \cdot \frac{k^{10}}{k^4} = 3k^{10-4} = 3k^6$.

We use the rule for dividing exponents with the same base:

$\frac{a^m}{a^n} = a^{m-n}$.

So the expression becomes $(3k^6)^3$.

Next, apply the power of a product rule and the power of a power rule:

$(ab)^n = a^n b^n$

$(a^m)^n = a^{m \cdot n}$.

Thus, $(3k^6)^3 = 3^3 (k^6)^3 = 27 k^{6 \cdot 3} = 27k^{18}$.

3. Final Answer

$27k^{18}$