To simplify the expression [tex]\(\left(x^{\frac{2}{5}}\right)^{\frac{5}{6}}\)[/tex], we need to use the properties of exponents, specifically the power of a power rule.
The power of a power rule states that [tex]\(\left(a^m\right)^n = a^{m \cdot n}\)[/tex].
Applying this rule to our expression:
[tex]\[
\left(x^{\frac{2}{5}}\right)^{\frac{5}{6}} = x^{\left(\frac{2}{5} \cdot \frac{5}{6}\right)}
\][/tex]
Now we multiply the exponents [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]:
[tex]\[
\frac{2}{5} \cdot \frac{5}{6} = \frac{2 \cdot 5}{5 \cdot 6} = \frac{10}{30} = \frac{1}{3}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
x^{\frac{1}{3}}
\][/tex]
Among the provided options:
- [tex]\(x^{\frac{37}{30}}\)[/tex]
- [tex]\(x^{\frac{13}{30}}\)[/tex]
- [tex]\(x^{\frac{10}{11}}\)[/tex]
- [tex]\(x^{\frac{1}{3}}\)[/tex]
The correct option is:
[tex]\[
\boxed{x^{\frac{1}{3}}}
\][/tex]
