To simplify the expression [tex]\(\left(8x^2\right)^3\)[/tex], we need to apply the rules of exponents and algebra. Let's break it down step by step:
1. Exponentiate the constant:
The constant inside the parentheses is 8. We need to raise 8 to the power of 3.
[tex]\[
8^3 = 8 \times 8 \times 8 = 512
\][/tex]
2. Exponentiate the variable:
The variable inside the parentheses is [tex]\(x^2\)[/tex]. We need to raise [tex]\(x^2\)[/tex] to the power of 3.
[tex]\[
(x^2)^3 = x^{2 \times 3} = x^6
\][/tex]
3. Combine the results:
Combining the two parts, we get:
[tex]\[
(8x^2)^3 = 512 \cdot x^6
\][/tex]
Therefore, the correct simplification of the expression [tex]\(\left(8x^2\right)^3\)[/tex] is [tex]\(512x^6\)[/tex].
So, the correct choice is [tex]\(\boxed{512x^6}\)[/tex].
