To rewrite the given expression [tex]\(\left(4^{\frac{2}{5}}\right)^{\frac{1}{4}}\)[/tex] with a rational exponent as a radical expression, we can follow these steps:
1. Understand the initial expression:
The given expression is:
[tex]\[
\left(4^{\frac{2}{5}}\right)^{\frac{1}{4}}
\][/tex]
2. Apply the power of a power rule:
According to the rules of exponents, [tex]\((a^{m})^{n} = a^{m \cdot n}\)[/tex]. So, we apply this rule here:
[tex]\[
\left(4^{\frac{2}{5}}\right)^{\frac{1}{4}} = 4^{\left(\frac{2}{5} \cdot \frac{1}{4}\right)}
\][/tex]
3. Simplify the exponent:
Simplify the product of the exponents [tex]\(\frac{2}{5} \cdot \frac{1}{4}\)[/tex]:
[tex]\[
\frac{2}{5} \cdot \frac{1}{4} = \frac{2 \cdot 1}{5 \cdot 4} = \frac{2}{20} = \frac{1}{10}
\][/tex]
4. Rewrite the expression with the simplified exponent:
So, the expression becomes:
[tex]\[
4^{\frac{1}{10}}
\][/tex]
5. Convert to a radical expression:
A rational exponent [tex]\(\frac{1}{n}\)[/tex] corresponds to the [tex]\(n\)[/tex]-th root. Therefore, [tex]\(\frac{1}{10}\)[/tex] corresponds to the 10th root:
[tex]\[
4^{\frac{1}{10}} = \sqrt[10]{4}
\][/tex]
In conclusion, the expression [tex]\(\left(4^{\frac{2}{5}}\right)^{\frac{1}{4}}\)[/tex] can be rewritten as:
[tex]\[
\sqrt[10]{4}
\][/tex]
So, the final answer is the 10th root of 4.
