Let's evaluate each equation one by one:
A. \( (2^2)^4 = 2^6 \)
Using the power of a power property, we have:
\( (a^m)^n = a^{m \cdot n} \)
So, \( (2^2)^4 = 2^{2 \cdot 4} = 2^8 \)
Thus, \( 2^8 \neq 2^6 \).
This equation is incorrect.
B. \( (3^5)^{-2} = \frac{1}{3^{-10}} \)
Again using the power of a power property:
\( (a^m)^{-n} = a^{-m \cdot n} \)
So, \( (3^5)^{-2} = 3^{-10} \)
Now, the right side \( \frac{1}{3^{-10}} = 3^{10} \).
Thus, \( 3^{-10} \neq 3^{10} \).
This equation is incorrect.
C. \( (5^{-3})^6 = \frac{1}{5^{18}} \)
Using the power of a power property again:
\( (5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18} \)
Since \( 5^{-18} = \frac{1}{5^{18}} \),
This equation is correct.
D. \( (8^3)^2 = 8^9 \)
Using the power of a power property:
\( (8^3)^2 = 8^{3 \cdot 2} = 8^6 \)
Thus, \( 8^6 \neq 8^9 \).
This equation is incorrect.
In conclusion, the only correct equation is: C. \( (5^{-3})^6 = \frac{1}{5^{18}} \).