To determine the correct expression for [tex]\(\left(\frac{2}{x}\right)^{\frac{1}{5}}\)[/tex], we can use the power of a quotient property. Let's proceed step by step to find the equivalent expression.
1. Understand the Power of a Quotient Rule:
The power of a quotient property states that:
[tex]\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\][/tex]
for any real numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] (where [tex]\(b \neq 0\)[/tex]) and any real number [tex]\(n\)[/tex].
2. Apply the Power of a Quotient Rule:
Given the expression [tex]\(\left(\frac{2}{x}\right)^{\frac{1}{5}}\)[/tex], we can apply the power of a quotient rule:
[tex]\[
\left(\frac{2}{x}\right)^{\frac{1}{5}} = \frac{2^{\frac{1}{5}}}{x^{\frac{1}{5}}}
\][/tex]
3. Identify the Correct Option:
Now we compare our simplified expression [tex]\(\frac{2^{\frac{1}{5}}}{x^{\frac{1}{5}}}\)[/tex] with the provided options:
- [tex]\(\frac{10}{x^{\frac{1}{b}}}\)[/tex]
- [tex]\(\frac{2^5}{x^5}\)[/tex]
- [tex]\(\frac{2^{\frac{1}{5}}}{x^{\frac{1}{6}}}\)[/tex]
- [tex]\(\frac{x^{\frac{1}{5}}}{2^{\frac{1}{5}}}\)[/tex]
The correct expression that matches our result is:
[tex]\[
\frac{2^{\frac{1}{5}}}{x^{\frac{1}{5}}}
\][/tex]
These correspond to the option [tex]\(\frac{2^{\frac{1}{5}}}{x^{\frac{1}{5}}}\)[/tex], and this is indeed the third option in the given list.
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{2^{\frac{1}{5}}}{x^{\frac{1}{5}}}}
\][/tex]
