Let's evaluate each equation one by one to determine which is correct.
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, \(2^8 \neq 2^6\).
This equation is incorrect.
B. \((3^5)^{-2} = \frac{1}{3^{-10}}\)
Using the power of a power rule:
\((3^5)^{-2} = 3^{5 \cdot -2} = 3^{-10}\)
Thus, \((3^5)^{-2} = 3^{-10}\), and \(\frac{1}{3^{-10}} = 3^{10}\).
So, \(3^{-10} \neq 3^{10}\).
This equation is incorrect.
C. \((5^3)^6 = \frac{1}{5^{18}}\)
Using the power of a power rule again:
\((5^3)^6 = 5^{3 \cdot 6} = 5^{18}\)
Thus, \((5^3)^6 = 5^{18}\), and \(\frac{1}{5^{18}} = 5^{-18}\).
So, \(5^{18} \neq \frac{1}{5^{18}}\).
This equation is incorrect.
D. \((8^3)^2 = 8^9\)
Using the power of a power rule:
\((8^3)^2 = 8^{3 \cdot 2} = 8^6\)
So, \(8^6 \neq 8^9\).
This equation is incorrect.
Upon evaluating all the equations, none of the given equations are correct.