To simplify the expression [tex]\(\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x}\)[/tex], we need to use the rules of exponents.
1. Recall that the fifth root of [tex]\(x\)[/tex] can be written as an exponent:
[tex]\[
\sqrt[5]{x} = x^{\frac{1}{5}}
\][/tex]
2. Given the expression, we now have:
[tex]\[
\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} = x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}}
\][/tex]
3. When multiplying expressions with the same base, we add the exponents. Therefore:
[tex]\[
x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} = x^{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}}
\][/tex]
4. Simplify the exponents by adding them together:
[tex]\[
\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5}
\][/tex]
So, the simplified form of [tex]\(\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x}\)[/tex] is:
[tex]\[
x^{\frac{4}{5}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
x^{\frac{4}{5}}
\][/tex]
