To solve the expression [tex]\((6i)^3\)[/tex], let's break it down step-by-step:
1. Identify the given expression: [tex]\((6i)^3\)[/tex].
2. Calculate the value of [tex]\((6i)^3\)[/tex]:
- [tex]\(i\)[/tex] is the imaginary unit, where [tex]\(i^2 = -1\)[/tex].
- First raise [tex]\(6i\)[/tex] to the power of 3:
[tex]\[
(6i)^3 = 6^3 \cdot i^3
\][/tex]
3. Compute [tex]\(6^3\)[/tex]:
[tex]\[
6^3 = 216
\][/tex]
4. Compute [tex]\(i^3\)[/tex]:
- Recall that [tex]\(i^3 = i \cdot i^2 = i \cdot (-1) = -i\)[/tex].
5. Combine the results:
- Substitute the values obtained:
[tex]\[
(6i)^3 = 216 \cdot (-i) = -216i
\][/tex]
Therefore, the expression [tex]\((6i)^3\)[/tex] is equivalent to [tex]\(-216i\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{-216i} \][/tex]
This matches option A.
