Sure, let's break this down step-by-step:
First, we start with the expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex].
1. Express each term using exponents:
[tex]\[
\sqrt[7]{x} = x^{\frac{1}{7}}
\][/tex]
Thus, our expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] can be written as:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
2. Combine the exponents:
When we multiply terms with the same base, we add the exponents. We have:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\left(\frac{1}{7} + \frac{1}{7} + \frac{1}{7}\right)}
\][/tex]
3. Add the exponents:
[tex]\[
\frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7}
\][/tex]
4. Write the final expression:
[tex]\[
x^{\frac{3}{7}}
\][/tex]
So, the simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(\boxed{x^{\frac{3}{7}}}\)[/tex].
