Step 1

Consider the expression \(\displaystyle\sqrt{{{x}\sqrt{{{x}\sqrt{{{x}}}}}}}\)

\(\displaystyle\sqrt{{{x}\sqrt{{{x}\sqrt{{{x}}}}}}}={\left({x}{\left({x}{\left({x}^{{\frac{{1}}{{2}}}}\right)}\right)}^{{\frac{{1}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}\)

Use the product property and the expression becomes,

\(\displaystyle{\left({x}{\left({x}{\left({x}^{{\frac{{1}}{{2}}}}\right)}\right)}^{{\frac{{1}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}={\left({x}{\left({x}^{{{1}+\frac{{1}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}\)

\(\displaystyle={\left({x}{\left({x}^{{\frac{{3}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}\right)}^{{\frac{{1}}{{2}}}}\)

\(\displaystyle{\left({x}\cdot\ {x}^{{\frac{{3}}{{4}}}}\right)}^{{\frac{{1}}{{2}}}}\)

Add the powers of the same bases.

\(\displaystyle{\left({x}\cdot\ {x}^{{\frac{{3}}{{4}}}}\right)}^{{\frac{{1}}{{2}}}}={\left({x}^{{{1}+\frac{{3}}{{4}}}}\right)}^{{\frac{{1}}{{2}}}}\)

\(\displaystyle={\left({x}^{{\frac{{7}}{{4}}}}\right)}^{{\frac{{1}}{{2}}}}\)

\(\displaystyle={x}^{{\frac{{7}}{{8}}}}\)

The obtained expression with rational exponents can be rewritten into radicals as.

\(\displaystyle{x}^{{\frac{{7}}{{8}}}}={\left({x}^{{{7}}}\right)}^{{\frac{{1}}{{8}}}}\)

\(=\sqrt[8]{x^{7}}\)

Therefore, the value of the expression is \(=\sqrt[8]{x^{7}}\)