Sure! Let's simplify the expression [tex]\(\sqrt[7]{x} \cdot 7^7 \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] step-by-step.
1. Expression Breakdown:
We start with [tex]\(\sqrt[7]{x} \cdot 7^7 \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex].
2. Rewrite the Radicals as Exponents:
Recall that [tex]\(\sqrt[7]{x}\)[/tex] can be written as [tex]\(x^{1/7}\)[/tex]. Thus, we can rewrite the expression as:
[tex]\[
x^{1/7} \cdot 7^7 \cdot x^{1/7} \cdot x^{1/7}
\][/tex]
3. Combine Like Terms:
By the properties of exponents, we can combine [tex]\(x^{1/7}\)[/tex] terms:
[tex]\[
x^{1/7} \cdot x^{1/7} \cdot x^{1/7} = x^{1/7 + 1/7 + 1/7} = x^{3/7}
\][/tex]
4. Substitute and Simplify:
Replacing back into the original expression, we get:
[tex]\[
7^7 \cdot x^{3/7}
\][/tex]
5. Final Simplified Form:
The final simplified form of the expression [tex]\(\sqrt[7]{x} \cdot 7^7 \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is:
[tex]\[
823543 \cdot x^{3/7}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
823543 \cdot x^{3/7}
\][/tex]
Therefore, the correct choice among the given options would be [tex]\(823543 \cdot x^{3/7}\)[/tex].
