To solve the given expression [tex]\(\sqrt[3]{a 12 b i s}\)[/tex], we need to find the cube roots of each term individually and then combine them. Let's break it down step-by-step:
1. Identify the terms:
- [tex]\(a\)[/tex] is an unspecified constant.
- 12 is a numerical constant.
- [tex]\(b\)[/tex] is an unspecified constant.
- [tex]\(i\)[/tex] is the imaginary unit, where [tex]\(i = \sqrt{-1}\)[/tex].
- [tex]\(s\)[/tex] is another unspecified constant.
2. Cube root of the numerical term 12:
The cube root of 12 is approximately [tex]\(2.2894284851066637\)[/tex].
3. Cube root of another numerical term 27:
The cube root of 27 is exactly [tex]\(3.0\)[/tex].
4. Cube root of the imaginary unit [tex]\(i\)[/tex]:
Finding the cube root of [tex]\(i\)[/tex] involves complex numbers. The cube root of [tex]\(i\)[/tex] is approximately [tex]\((0.8660254037844387 + 0.49999999999999994i)\)[/tex].
5. Cube root of [tex]\(s\)[/tex] with the assumption that it's 1 (if not specified otherwise):
The cube root of 1 is exactly [tex]\(1.0\)[/tex].
Putting these values together, we get:
- Cube root of 12 is [tex]\(2.2894284851066637\)[/tex].
- Cube root of 27 is [tex]\(3.0\)[/tex].
- Cube root of [tex]\(i\)[/tex] is [tex]\((0.8660254037844387 + 0.49999999999999994i)\)[/tex].
- Cube root of [tex]\(s\)[/tex] is [tex]\(1.0\)[/tex].
Thus, the solution to the expression [tex]\(\sqrt[3]{a 12 b i s}\)[/tex] is:
[tex]\[
\left(2.2894284851066637, 3.0, (0.8660254037844387+0.49999999999999994i), 1.0\right)
\][/tex]
