To rewrite [tex]\(8^{\frac{1}{2}}\)[/tex] using a root, we need to understand the relationship between exponents and roots.
The expression [tex]\(8^{\frac{1}{2}}\)[/tex] is asking for 8 raised to the power of [tex]\( \frac{1}{2} \)[/tex]. According to the rules of exponents, raising a number to the power of [tex]\( \frac{1}{2} \)[/tex] is equivalent to taking the square root of that number.
Therefore, [tex]\(8^{\frac{1}{2}}\)[/tex] can be rewritten as [tex]\( \sqrt{8} \)[/tex].
Let's examine the given options:
1. [tex]\( \sqrt[8]{\frac{1}{2}} \)[/tex] - This represents the eighth root of [tex]\( \frac{1}{2} \)[/tex], which is not what we need.
2. [tex]\( \sqrt[\frac{1}{2}]{8} \)[/tex] - This represents taking the [tex]\(\frac{1}{2}\)[/tex]-th root of 8, which is not correct either.
3. [tex]\( \sqrt[1]{8} \)[/tex] - This represents the first root of 8, which is simply 8.
4. [tex]\( \sqrt{8} \)[/tex] - This represents the square root of 8.
The correct option that rewrites [tex]\(8^{\frac{1}{2}}\)[/tex] using a root is [tex]\( \sqrt{8} \)[/tex].
So, the answer is option 4.
