To convert the expression [tex]\(6^{\frac{1}{12}}\)[/tex] into radical form, we need to recall a fundamental principle of exponents and radicals. In general, if we have an expression of the form [tex]\(x^{\frac{1}{n}}\)[/tex], it can be rewritten as the [tex]\(n\)[/tex]-th root of [tex]\(x\)[/tex], which is [tex]\(\sqrt[n]{x}\)[/tex].
Given:
[tex]\[ 6^{\frac{1}{12}} \][/tex]
We can rewrite this using the relationship between exponents and radicals:
[tex]\[ 6^{\frac{1}{12}} = \sqrt[12]{6} \][/tex]
None of the provided answer choices directly match this form. Let's analyze each option briefly to confirm:
1. [tex]\(\sqrt[7]{6^{12}}\)[/tex]:
- This is the 7th root of [tex]\(6^{12}\)[/tex], which is not equivalent to [tex]\(6^{\frac{1}{12}}\)[/tex].
2. [tex]\(\sqrt[12]{6^7}\)[/tex]:
- This is the 12th root of [tex]\(6^7\)[/tex], which is not equivalent to [tex]\(6^{\frac{1}{12}}\)[/tex].
3. [tex]\(\sqrt[12]{7 \cdot 6}\)[/tex]:
- This is the 12th root of the product [tex]\(7 \cdot 6\)[/tex], which is also not equivalent to [tex]\(6^{\frac{1}{12}}\)[/tex].
Thus, the correct radical form for [tex]\(6^{\frac{1}{12}}\)[/tex] is:
[tex]\[ \sqrt[12]{6} \][/tex]
