To rewrite the radical expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form, let's follow these steps meticulously:
1. Identify the Radical Expression:
You are given:
[tex]\[
(\sqrt{x})^5
\][/tex]
2. Convert the Radical to a Rational Exponent:
Recall that the square root of [tex]\(x\)[/tex] can be represented as a rational exponent. The square root of [tex]\(x\)[/tex] is equivalent to [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{2}\)[/tex]. Therefore:
[tex]\[
\sqrt{x} = x^{\frac{1}{2}}
\][/tex]
3. Raise the Equivalent Expression to the Given Power:
The given expression raises [tex]\(\sqrt{x}\)[/tex] to the 5th power. Substituting the rational exponent form of [tex]\(\sqrt{x}\)[/tex], we get:
[tex]\[
(\sqrt{x})^5 = (x^{\frac{1}{2}})^5
\][/tex]
4. Apply the Power Rule for Exponents:
When raising a power to another power, you multiply the exponents. Here, we need to multiply [tex]\(\frac{1}{2}\)[/tex] by 5:
[tex]\[
(x^{\frac{1}{2}})^5 = x^{\frac{1}{2} \times 5}
\][/tex]
Solving the multiplication in the exponent:
[tex]\[
\frac{1}{2} \times 5 = \frac{5}{2}
\][/tex]
5. Write the Final Expression:
Therefore, the expression [tex]\((\sqrt{x})^5\)[/tex] in rational exponent form is:
[tex]\[
x^{\frac{5}{2}}
\][/tex]
Based on this detailed step-by-step solution, the correct answer is:
A. [tex]\(x^{\frac{5}{2}}\)[/tex]
