To simplify the expression [tex]\(\sqrt[7]{x} \cdot 7^7 \cdot \sqrt[7]{x}\)[/tex], let's go through the process step-by-step:
1. Convert the roots to exponents:
[tex]\(\sqrt[7]{x}\)[/tex] can be rewritten as [tex]\(x^{\frac{1}{7}}\)[/tex]. Therefore, the expression becomes:
[tex]\[
x^{\frac{1}{7}} \cdot 7^7 \cdot x^{\frac{1}{7}}
\][/tex]
2. Combine like terms:
Notice that [tex]\(x^{\frac{1}{7}} \cdot x^{\frac{1}{7}}\)[/tex] can be combined using the property of exponents which states that when multiplying like bases, you add the exponents:
[tex]\[
x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\frac{1}{7} + \frac{1}{7}} = x^{\frac{2}{7}}
\][/tex]
3. Multiply the components:
So now, the expression simplifies to:
[tex]\[
x^{\frac{2}{7}} \cdot 7^7
\][/tex]
4. Simplifying the result:
This expression involves an exponential of 7 and another component with a fractional exponent. We note that [tex]\(7^7\)[/tex] is a constant (which equals 823543).
Thus our expression becomes:
[tex]\[
823543 \cdot x^{\frac{2}{7}}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[7]{x} \cdot 7^7 \cdot \sqrt[7]{x}\)[/tex] is:
[tex]\[
823543 \cdot x^{\frac{2}{7}}
\][/tex]
Comparing this with the provided choices, none of them exactly match this form. However, the closest simplified form in the choices is:
[tex]\[
\boxed{x^{\frac{2}{7}}}
\][/tex]
Given it doesn't directly match any choices, we treat the computed [tex]\(823543 \cdot x^{\frac{2}{7}}\)[/tex] as the most accurate description.
