The answer provided by the student is incorrect. Let's go through the steps of simplifying the given rational expression correctly:
The given expression is:
[tex]\[
\left(\frac{x^{\frac{2}{5}} \cdot x^{\frac{4}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}
\][/tex]
1. Combine the Exponents in the Numerator:
Using the property of exponents that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we combine the exponents in the numerator:
[tex]\[
x^{\frac{2}{5}} \cdot x^{\frac{4}{5}} = x^{\frac{2}{5} + \frac{4}{5}} = x^{\frac{6}{5}}
\][/tex]
Therefore, the expression now is:
[tex]\[
\left(\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}
\][/tex]
2. Simplify the Fraction:
Using the property of exponents that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we simplify the fraction:
[tex]\[
\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}} = x^{\frac{6}{5} - \frac{2}{5}} = x^{\frac{4}{5}}
\][/tex]
Now the expression is:
[tex]\[
\left(x^{\frac{4}{5}}\right)^{\frac{1}{2}}
\][/tex]
3. Apply the Power of a Power Rule:
Using the property of exponents that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we apply the power of a power rule:
[tex]\[
\left(x^{\frac{4}{5}}\right)^{\frac{1}{2}} = x^{\frac{4}{5} \cdot \frac{1}{2}} = x^{\frac{4}{10}} = x^{\frac{2}{5}}
\][/tex]
Therefore, the correct simplified form of the given rational expression is:
[tex]\[
x^{\frac{2}{5}}
\][/tex]
The student's final result of [tex]\(x^{\frac{3}{2}}\)[/tex] was incorrect. The correct simplified form is [tex]\(x^{\frac{2}{5}}\)[/tex].
