To simplify the expression [tex]\(\left(x^{\frac{2}{5}}\right)^{10}\)[/tex], let's use the properties of exponents. Specifically, we will use the power of a power rule which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
Here is the step-by-step process:
1. Identify the base and the exponents:
[tex]\[
\text{Base: } x
\][/tex]
[tex]\[
\text{First exponent: } \frac{2}{5}
\][/tex]
[tex]\[
\text{Second exponent: } 10
\][/tex]
2. Apply the power of a power rule:
[tex]\[
\left(x^{\frac{2}{5}}\right)^{10} = x^{\left(\frac{2}{5} \cdot 10\right)}
\][/tex]
3. Perform the multiplication in the exponent:
[tex]\[
\frac{2}{5} \cdot 10 = \left(\frac{2 \cdot 10}{5}\right)
\][/tex]
[tex]\[
= \frac{20}{5}
\][/tex]
[tex]\[
= 4
\][/tex]
4. Write the simplified expression:
[tex]\[
x^4
\][/tex]
So, [tex]\(\left(x^{\frac{2}{5}}\right)^{10}\)[/tex] simplifies to [tex]\(x^4\)[/tex].
Therefore, among the given choices [tex]\(x^{\frac{1}{5}}, x^{\frac{1}{4}}, x^4, x^5\)[/tex], the correct answer is:
[tex]\[
x^4
\][/tex]
