The student's answer is not correct. Let's go through the correct simplification step by step.
1. Start with the given expression:
[tex]\[
\left(\frac{x^{\frac{2}{5}} \cdot x^{\frac{4}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}
\][/tex]
2. Combine the exponents in the numerator:
[tex]\[
x^{\frac{2}{5}} \cdot x^{\frac{4}{5}} = x^{\left(\frac{2}{5} + \frac{4}{5}\right)} = x^{\frac{6}{5}}
\][/tex]
Now the expression is:
[tex]\[
\left(\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}
\][/tex]
3. Simplify the fraction by subtracting the exponents (using the property [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]):
[tex]\[
\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}} = x^{\left(\frac{6}{5} - \frac{2}{5}\right)} = x^{\frac{4}{5}}
\][/tex]
Now we have:
[tex]\[
\left(x^{\frac{4}{5}}\right)^{\frac{1}{2}}
\][/tex]
4. Take the square root (or equivalently, raise the expression to the power of [tex]\( \frac{1}{2} \)[/tex]):
[tex]\[
\left(x^{\frac{4}{5}}\right)^{\frac{1}{2}} = x^{\left(\frac{4}{5} \cdot \frac{1}{2}\right)} = x^{\frac{4}{10}} = x^{\frac{2}{5}}
\][/tex]
Therefore, the correct simplified form of the given expression is:
[tex]\[
x^{\frac{2}{5}}
\][/tex]
The student's error occurred in the final step where they incorrectly simplified [tex]\(\left(x^{\frac{4}{5}}\right)^{\frac{1}{2}}\)[/tex] to [tex]\(x^{\frac{3}{2}}\)[/tex] instead of the correct [tex]\(x^{\frac{2}{5}}\)[/tex].
