To simplify the given expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], let's follow a detailed, step-by-step approach:
1. Understand the Radicals:
- The radical expression [tex]\(\sqrt[7]{x}\)[/tex] can be written in exponential form as [tex]\(x^{\frac{1}{7}}\)[/tex]. This is because the [tex]\(n\)[/tex]-th root of [tex]\(x\)[/tex], [tex]\(\sqrt[n]{x}\)[/tex], is equivalent to [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{n}\)[/tex].
2. Rewrite the Expression Using Exponents:
- We now need to simplify [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex].
- Converting the radicals to exponents, the expression becomes:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}
\][/tex]
3. Apply the Rules of Exponents:
- When multiplying expressions with the same base, you add the exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Therefore, adding the exponents in our expression:
[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]
4. Add the Exponents:
- Sum up the exponents:
[tex]\[
\frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7}
\][/tex]
5. Write the Simplified Form:
- The simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(x^{\frac{3}{7}}\)[/tex].
Thus, the correct simplified form is [tex]\(\boxed{x^{\frac{3}{7}}}\)[/tex].
