Sure, let's verify the correctness of each equation step by step.
### Equation A: [tex]\(\left(2^2\right)^4 = 2^6\)[/tex]
1. Calculate the left-hand side (LHS):
[tex]\[
\left(2^2\right)^4 = 4^4 = 256
\][/tex]
2. Calculate the right-hand side (RHS):
[tex]\[
2^6 = 64
\][/tex]
3. Compare the LHS and RHS:
[tex]\[
256 \neq 64
\][/tex]
Therefore, Equation A is incorrect.
### Equation B: [tex]\(\left(3^5\right)^{-2} = \frac{1}{3^{-10}}\)[/tex]
1. Calculate the LHS:
[tex]\[
\left(3^5\right)^{-2} = 3^{5 \cdot (-2)} = 3^{-10}
\][/tex]
2. Calculate the RHS:
[tex]\[
\frac{1}{3^{-10}} = 3^{10} \quad (\text{since} \; a^{-n} = \frac{1}{a^n})
\][/tex]
3. Compare the LHS and RHS:
[tex]\[
3^{-10} \neq 3^{10}
\][/tex]
Therefore, Equation B is incorrect.
### Equation C: [tex]\(\left(5^{-3}\right)^6 = \frac{1}{5^{18}}\)[/tex]
1. Calculate the LHS:
[tex]\[
\left(5^{-3}\right)^6 = 5^{-3 \cdot 6} = 5^{-18}
\][/tex]
2. Calculate the RHS:
[tex]\[
\frac{1}{5^{18}} = 5^{-18} \quad (\text{since} \; \frac{1}{a^n} = a^{-n})
\][/tex]
3. Compare the LHS and RHS:
[tex]\[
5^{-18} = 5^{-18}
\][/tex]
Therefore, Equation C is correct.
### Equation D: [tex]\(\left(8^3\right)^2 = 8^9\)[/tex]
1. Calculate the LHS:
[tex]\[
\left(8^3\right)^2 = 8^{3 \cdot 2} = 8^6
\][/tex]
2. Calculate the RHS:
[tex]\[
8^9
\][/tex]
3. Compare the LHS and RHS:
[tex]\[
8^6 \neq 8^9
\][/tex]
Therefore, Equation D is incorrect.
In conclusion, the correct equation is:
[tex]\[
\boxed{C}
\][/tex]
