Let's evaluate each of the equations one by one:
A. \((8^3)^2 = 8^9\)
Using the power of a power property \( (a^m)^n = a^{m \cdot n} \): \((8^3)^2 = 8^{3 \cdot 2} = 8^6\).
This is not equal to \(8^9\). Therefore, A is incorrect.
B. \((2^2)^4 = 2^6\)
Applying the same property: \((2^2)^4 = 2^{2 \cdot 4} = 2^8\).
This is not equal to \(2^6\). Therefore, B is incorrect.
C. \((5^{-3})^6 = \frac{1}{5^{18}}\)
Using the power of a power property: \((5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}\).
To express this as a fraction: \(5^{-18} = \frac{1}{5^{18}}\).
Therefore, C is correct.
D. \((3^5)^{-2} = \frac{1}{3^{-10}}\)
Using the property: \((3^5)^{-2} = 3^{5 \cdot -2} = 3^{-10}\). But \(\frac{1}{3^{-10}} = 3^{10}\), which is not equal to \(3^{-10}\).
Therefore, D is incorrect.
The only correct equation is:
C. \((5^{-3})^6 = \frac{1}{5^{18}}\).