Let's evaluate each equation one by one.
A. \((2^2)^4 = 2^6\)
Using the property \((a^m)^n = a^{m \cdot n}\):
\((2^2)^4 = 2^{2 \cdot 4} = 2^8\),
which is not equal to \(2^6\).
This equation is incorrect.
B. \((3^5)^{-2} = 1/3^{-10}\)
Using the property \(a^{-n} = \frac{1}{a^n}\):
\((3^5)^{-2} = \frac{1}{(3^5)^2} = \frac{1}{3^{5 \cdot 2}} = \frac{1}{3^{10}}\).
On the right side, \(1/3^{-10} = 3^{10}\), as \(1/a^{-n} = a^{n}\).
This equation is incorrect.
C. \((5^{-3})^6 = 1/5^{18}\)
Using the property \((a^m)^n = a^{m \cdot n}\):
\((5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}\).
Using \(5^{-18} = \frac{1}{5^{18}}\), we have equality:
\((5^{-3})^6 = 1/5^{18}\).
This equation is correct.
D. \((8^3)^ = 8^9\)
It seems there might be a typographical error or missing exponent in the expression \((8^3)^ = 8^9\). The left side should have a valid exponent. Assuming it's meant to be \((8^3)^3\), we can evaluate that:
Using \( (a^m)^n = a^{m \cdot n} \):
\((8^3)^3 = 8^{3 \cdot 3} = 8^9\).
However, without proper formatting or context of the equation, we cannot fully evaluate it. As it stands, it can't be declared correct.
This equation lacks clarity, but if corrected might be accurate.
In conclusion, the only clearly correct option is C. \((5^{-3})^6 = 1/5^{18}\).