To rewrite the radical expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form, let's follow these steps:
1. Understand the radical notation: The expression [tex]\(\sqrt{x}\)[/tex] can be rewritten in exponent notation as [tex]\(x^{\frac{1}{2}}\)[/tex]. This is because the square root of [tex]\(x\)[/tex] is the same as raising [tex]\(x\)[/tex] to the power of [tex]\(\frac{1}{2}\)[/tex].
2. Apply exponent rules: The given expression is [tex]\((\sqrt{x})^5\)[/tex]. Substituting [tex]\(\sqrt{x}\)[/tex] with [tex]\(x^{\frac{1}{2}}\)[/tex], we get [tex]\((x^{\frac{1}{2}})^5\)[/tex].
3. Simplify using exponent multiplication rule: According to the properties of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this rule, [tex]\((x^{\frac{1}{2}})^5\)[/tex] becomes [tex]\(x^{\frac{1}{2} \cdot 5}\)[/tex].
4. Calculate the exponent: Multiply [tex]\(\frac{1}{2}\)[/tex] by 5 to get [tex]\(\frac{5}{2}\)[/tex]. Therefore, [tex]\((\sqrt{x})^5\)[/tex] is equivalent to [tex]\(x^{\frac{5}{2}}\)[/tex].
Thus, the correct answer is:
[tex]\[
E. x^{\frac{5}{2}}
\][/tex]
It seems there is a typo in the available options, as none of them matches [tex]\(x^{\frac{5}{2}}\)[/tex]. Please ensure the options are verified or reevaluate the question context.
