The problem is to simplify the expression $(\frac{c^7}{-9d^4})^2$.

1. Problem Description

The problem is to simplify the expression $(\frac{c^7}{-9d^4})^2$.

2. Solution Steps

To simplify the expression, we need to apply the power of a quotient rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$.

Applying this rule, we have:

$(\frac{c^7}{-9d^4})^2 = \frac{(c^7)^2}{(-9d^4)^2}$

Next, we apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$.

$(c^7)^2 = c^{7 \cdot 2} = c^{14}$

Now, let's simplify the denominator.

$(-9d^4)^2 = (-9)^2 \cdot (d^4)^2 = 81 \cdot d^{4 \cdot 2} = 81d^8$

Therefore, the simplified expression is:

$\frac{c^{14}}{81d^8}$

3. Final Answer

$\frac{c^{14}}{81d^8}$