Let's break down the given mathematical expression step-by-step to simplify it:
[tex]\[
\sqrt[5]{x \sqrt[4]{x^3 \sqrt[3]{x^2 \sqrt{x}}}}
\][/tex]
1. Start with the innermost part: [tex]\(\sqrt{x}\)[/tex]:
[tex]\[ \sqrt{x} = x^{1/2} \][/tex]
2. Substitute [tex]\(\sqrt{x}\)[/tex] into the next part:
[tex]\[ \sqrt[3]{x^2 \sqrt{x}} = \sqrt[3]{x^2 \cdot x^{1/2}} \][/tex]
[tex]\[ = \sqrt[3]{x^{2 + 1/2}} \][/tex]
[tex]\[ = \sqrt[3]{x^{5/2}} \][/tex]
[tex]\[ = (x^{5/2})^{1/3} \][/tex]
[tex]\[ = x^{(5/2) \cdot 1/3} \][/tex]
[tex]\[ = x^{5/6} \][/tex]
3. Substitute [tex]\(x^{5/6}\)[/tex] into the next part:
[tex]\[ \sqrt[4]{x^3 \sqrt[3]{x^2 \sqrt{x}}} = \sqrt[4]{x^3 \cdot x^{5/6}} \][/tex]
[tex]\[ = \sqrt[4]{x^{3 + 5/6}} \][/tex]
[tex]\[ = \sqrt[4]{x^{18/6 + 5/6}} \][/tex]
[tex]\[ = \sqrt[4]{x^{23/6}} \][/tex]
[tex]\[ = (x^{23/6})^{1/4} \][/tex]
[tex]\[ = x^{(23/6) \cdot 1/4} \][/tex]
[tex]\[ = x^{23/24} \][/tex]
4. Finally, substitute [tex]\(x^{23/24}\)[/tex] into the outermost part:
[tex]\[ \sqrt[5]{x \sqrt[4]{x^3 \sqrt[3]{x^2 \sqrt{x}}}} = \sqrt[5]{x \cdot x^{23/24}} \][/tex]
[tex]\[ = \sqrt[5]{x^{1 + 23/24}} \][/tex]
[tex]\[ = \sqrt[5]{x^{24/24 + 23/24}} \][/tex]
[tex]\[ = \sqrt[5]{x^{47/24}} \][/tex]
[tex]\[ = (x^{47/24})^{1/5} \][/tex]
[tex]\[ = x^{(47/24) \cdot 1/5} \][/tex]
[tex]\[ = x^{47/120} \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
x^{47/120}
\][/tex]
