Certainly! Let's rewrite the given radical expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form. We'll go through it step-by-step:
1. Understand the Square Root:
The square root of [tex]\(x\)[/tex], [tex]\(\sqrt{x}\)[/tex], can be expressed using a rational exponent as [tex]\(x^{\frac{1}{2}}\)[/tex].
2. Rewrite the Expression:
We substitute [tex]\(\sqrt{x}\)[/tex] with [tex]\(x^{\frac{1}{2}}\)[/tex]. This gives us:
[tex]\[
(\sqrt{x})^5 = (x^{\frac{1}{2}})^5
\][/tex]
3. Apply the Power of a Power Property:
When you have an expression [tex]\((a^m)^n\)[/tex], it simplifies to [tex]\(a^{m \cdot n}\)[/tex]. Applying this property:
[tex]\[
(x^{\frac{1}{2}})^5 = x^{\frac{1}{2} \cdot 5}
\][/tex]
4. Simplify the Exponent:
Multiply the exponents:
[tex]\[
\frac{1}{2} \cdot 5 = \frac{5}{2}
\][/tex]
So, we get:
[tex]\[
x^{\frac{5}{2}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x^{\frac{5}{2}}}
\][/tex]
