Let's rewrite the radical expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form step-by-step.
1. Understanding the Square Root:
The square root of [tex]\(x\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex]. Therefore, [tex]\(\sqrt{x} = x^{1/2}\)[/tex].
2. Applying the Property of Exponents:
Now, we need to deal with the given expression [tex]\((\sqrt{x})^5\)[/tex]. Substituting the square root in rational exponent form:
[tex]\[
(\sqrt{x})^5 = (x^{1/2})^5
\][/tex]
3. Using the Power of a Power Property:
When we raise a power to another power, we multiply the exponents. The property [tex]\((a^m)^n = a^{mn}\)[/tex] applies here.
[tex]\[
(x^{1/2})^5 = x^{(1/2) \cdot 5}
\][/tex]
4. Multiplying the Exponents:
Multiply [tex]\(\frac{1}{2}\)[/tex] by 5 to simplify the exponent.
[tex]\[
x^{(1/2) \cdot 5} = x^{5/2}
\][/tex]
Therefore, the expression [tex]\((\sqrt{x})^5\)[/tex] rewritten in rational exponent form is:
[tex]\[
x^{\frac{5}{2}}
\][/tex]
Since we have correctly identified the expression, the correct answer is:
[tex]\[
\boxed{x^{\frac{5}{2}}}
\][/tex]
Hence, the corresponding option is:
[tex]\[
\boxed{C}
\][/tex]
