To rewrite the expression [tex]\(8^{\frac{1}{2}}\)[/tex] using a root, we need to understand the relationship between exponents and roots. The exponent [tex]\(\frac{1}{2}\)[/tex] is equivalent to taking the square root of the base number.
Let's look at the options provided:
1. [tex]\(\sqrt[1]{8}\)[/tex] - This represents taking the 1st root of 8, which is just 8 itself. This is not equivalent to [tex]\(8^{\frac{1}{2}}\)[/tex].
2. [tex]\(\sqrt[8]{\frac{1}{2}}\)[/tex] - This represents taking the 8th root of [tex]\(\frac{1}{2}\)[/tex], which is not related to our original expression.
3. [tex]\(\sqrt[\frac{1}{2}]{8}\)[/tex] - This represents taking the root with an index of [tex]\(\frac{1}{2}\)[/tex], which is not a common way to express roots and is not equivalent to [tex]\(8^{\frac{1}{2}}\)[/tex].
4. [tex]\(\sqrt{8}\)[/tex] - This represents the principal square root of 8, which is exactly what [tex]\(8^{\frac{1}{2}}\)[/tex] signifies.
Thus, the correct way to rewrite [tex]\(8^{\frac{1}{2}}\)[/tex] using a root is [tex]\(\sqrt{8}\)[/tex]. This matches the correct representation and simplifies correctly to approximately 2.8284271247461903 as calculated. So, the correct choice is:
[tex]\(\boxed{\sqrt{8}}\)[/tex]
