To simplify the expression [tex]\({ }^7 \sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], let's go through it step-by-step.
1. Rewrite each term in exponent form:
- [tex]\({ }^7 \sqrt{x}\)[/tex] is the same as [tex]\(x^{\frac{1}{7}}\)[/tex]
- [tex]\(\sqrt[7]{x}\)[/tex] is also [tex]\(x^{\frac{1}{7}}\)[/tex]
- The second [tex]\(\sqrt[7]{x}\)[/tex] is similarly [tex]\(x^{\frac{1}{7}}\)[/tex]
Thus, the original expression is [tex]\(x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}\)[/tex].
2. Combine the exponents:
- Recall the property of exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
- Apply this property to 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]
- Add the exponents:
[tex]\[
\frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7}
\][/tex]
3. Simplified form:
[tex]\[
x^{\frac{3}{7}}
\][/tex]
Thus, the simplified form of [tex]\({ }^7 \sqrt{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(\boxed{x^{\frac{3}{7}}}\)[/tex].
