Let's evaluate each equation one by one:
A. \((2^{2})^{4} = 2^{6}\)
Using the power of a power rule, \((a^{m})^{n} = a^{m \cdot n}\): \((2^{2})^{4} = 2^{2 \cdot 4} = 2^{8}\)
So, \((2^{2})^{4} \neq 2^{6}\).
A is incorrect.
B. \((3^{5})^{-2} = \frac{1}{3^{-10}}\)
Using the power of a power rule: \((3^{5})^{-2} = 3^{5 \cdot (-2)} = 3^{-10}\)
On the right, \(\frac{1}{3^{-10}} = 3^{10}\).
So, \(3^{-10} \neq 3^{10}\).
B is incorrect.
C. \((5^{-3})^{6} = \frac{1}{5^{18}}\)
Using the power of a power rule: \((5^{-3})^{6} = 5^{-3 \cdot 6} = 5^{-18}\)
On the right, \(\frac{1}{5^{18}} = 5^{-18}\).
So, \(5^{-18} = 5^{-18}\).
C is correct.
D. \((8^{3})^{2} = 8^{9}\)
Using the power of a power rule: \((8^{3})^{2} = 8^{3 \cdot 2} = 8^{6}\)
So, \((8^{3})^{2} \neq 8^{9}\).
D is incorrect.
The only correct equation is C: \((5^{-3})^{6} = \frac{1}{5^{18}}\).