To simplify the expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], we will use the properties of exponents.
1. First, recognize that [tex]\(\sqrt[7]{x}\)[/tex] can be written as [tex]\(x^{\frac{1}{7}}\)[/tex].
2. Rewrite the given expression using this exponent form:
[tex]\[
\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} = x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
3. Next, use the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Add the exponents together:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}}
\][/tex]
4. Simplify the sum of the exponents:
[tex]\[
\frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{4}{7}
\][/tex]
5. Therefore, the expression simplifies to:
[tex]\[
x^{\frac{4}{7}}
\][/tex]
So, the simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(x^{\frac{4}{7}}\)[/tex].
The correct answer is:
[tex]\[
x^{\frac{4}{7}}
\][/tex]
