To simplify the expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], we start by using the property of exponents involving fractional powers.
First, recall that [tex]\(\sqrt[7]{x}\)[/tex] can be written as [tex]\(x^{\frac{1}{7}}\)[/tex]. Therefore, the given expression can be rewritten with exponents:
[tex]\[
\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} = x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
Next, we use the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Applying this property, we combine the exponents:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\frac{1}{7} + \frac{1}{7} + \frac{1}{7}}
\][/tex]
Add the exponents together:
[tex]\[
\frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7}
\][/tex]
So the simplified form of the expression is:
[tex]\[
x^{\frac{3}{7}}
\][/tex]
Therefore, the correct choice is:
[tex]\[
x^{\frac{3}{7}}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is indeed [tex]\(x^{\frac{3}{7}}\)[/tex]. This corresponds to the first choice in the provided list.
