Let's evaluate each equation one by one to determine which one is correct.
A. \((2^2)^4 = 2^6\)
Using the power of a power property \((a^m)^n = a^{mn}\):
\[ (2^2)^4 = 2^{2 \cdot 4} = 2^8 \]
So, the left side becomes \(2^8\), and the right side is \(2^6\).
This is incorrect since \(2^8 \neq 2^6\).
B. \((3^5)^{-2} = 1/3^{-10}\)
Starting with the left side:
\[ (3^5)^{-2} = 3^{5 \cdot -2} = 3^{-10} \]
For the right side, using \(1/a^{-n} = a^n\):
\[ 1/3^{-10} = 3^{10} \]
This means \(3^{-10} \neq 3^{10}\).
This is incorrect since \(3^{-10} \neq 3^{10}\).
C. \((5^{-3})^6 = 1/5^{18}\)
Evaluating the left side:
\[ (5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18} \]
For the right side, we use the property of exponents:
\[ 1/5^{18} = 5^{-18} \]
So, we have \(5^{-18} = 5^{-18}\).
This is correct.
D. \((8^3)^2 = 8^9\)
Using the power of a power property again:
\[ (8^3)^2 = 8^{3 \cdot 2} = 8^6 \]
So, the left side becomes \(8^6\) and the right side is \(8^9\).
This is incorrect since \(8^6 \neq 8^9\).
The correct equation is C. \((5^{-3})^6 = 1/5^{18}\).