Certainly! To address the question of rewriting [tex]\(8^{\frac{1}{2}}\)[/tex] using a root, let's go through the steps in detail.
1. Understanding Rational Exponents:
- A rational exponent in the form [tex]\( a^{\frac{m}{n}} \)[/tex] can be translated into a root. Specifically, [tex]\( a^{\frac{m}{n}} \)[/tex] means the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] raised to the power [tex]\( m \)[/tex].
2. Specific Case for [tex]\( 8^{\frac{1}{2}} \)[/tex]:
- Here, we have [tex]\( 8^{\frac{1}{2}} \)[/tex]. This notation indicates that we are dealing with the [tex]\( \frac{1}{2} \)[/tex]-th power of 8.
- According to the property of rational exponents, [tex]\( 8^{\frac{1}{2}} \)[/tex] implies the square root of 8, because a [tex]\( \frac{1}{2} \)[/tex]-th power is equivalent to a square root (since [tex]\( \frac{1}{2} \)[/tex] means 2 in the denominator of the exponent indicates a square root).
3. Translation to Root Form:
- Hence, [tex]\( 8^{\frac{1}{2}} \)[/tex] is equivalent to [tex]\( \sqrt{8} \)[/tex].
4. Conclusion:
- Among the given options, the correct way to rewrite [tex]\( 8^{\frac{1}{2}} \)[/tex] using a root is [tex]\( \sqrt{8} \)[/tex].
So, the correct option is:
[tex]\[ \sqrt{8} \][/tex]
This matches the choice listed in the options. Thus, we have successfully rewritten the given expression [tex]\( 8^{\frac{1}{2}} \)[/tex] as [tex]\( \sqrt{8} \)[/tex].
