Certainly! Let's rewrite the given radical expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form step-by-step.
1. Understand the Square Root as an Exponent:
The square root of [tex]\( x \)[/tex] can be written as [tex]\( x \)[/tex] raised to the power of [tex]\( \frac{1}{2} \)[/tex]. Therefore,
[tex]\[
\sqrt{x} = x^{\frac{1}{2}}
\][/tex]
2. Apply the Exponent to the Entire Expression:
Now we have the expression [tex]\((x^{\frac{1}{2}})^5\)[/tex].
3. Use the Property of Exponents:
When you raise a power to another power, you multiply the exponents. Here’s the rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this rule:
[tex]\[
(x^{\frac{1}{2}})^5 = x^{\frac{1}{2} \cdot 5}
\][/tex]
4. Multiply the Exponents:
Now, multiply the two exponents:
[tex]\[
\frac{1}{2} \cdot 5 = \frac{5}{2}
\][/tex]
5. Rewrite the Expression:
So, [tex]\((x^{\frac{1}{2}})^5\)[/tex] becomes:
[tex]\[
x^{\frac{5}{2}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B: x^{\frac{5}{2}}}
\][/tex]
