Step 1

Formula used:

Product property of radicals:

If a and b are real numbers and \(\displaystyle{n}{>}{1}\) is an integer, the product property is true provided that the radicals are real numbers.

\(\sqrt[n]{a^{m}}=a^{m/n}\)

If m and n are integers and \(\displaystyle{n}{>}{1}\) is an integer, then

\(\sqrt[n]{a^{m}}=a^{m/n}\)

Step 2

Consider the expression \(\sqrt[3]{y\sqrt[3]{y\sqrt[3]{y}}}\)

Rewrite the provided expression as rational exponents.

\(\sqrt[3]{y\sqrt[3]{y\sqrt[3]{y}}}={\left({y}{\left({y}{\left({y}^{{\frac{{1}}{{3}}}}\right)}\right)}^{{\frac{{1}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}\)

Use the product property and work from inside the parentheses and the expression becomes,

\(\displaystyle{\left({y}{\left({y}{\left({y}^{{\frac{{1}}{{3}}}}\right)}\right)}^{{\frac{{1}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}={\left({y}{\left({y}^{{{1}+\frac{{1}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}\)

\(\displaystyle={\left({y}{\left({y}^{{\frac{{4}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}\right)}^{{\frac{{1}}{{3}}}}\)

\(\displaystyle={\left({y}\cdot\ {y}^{{\frac{{4}}{{9}}}}\right)}^{{\frac{{1}}{{3}}}}\)

Add the powers of the same bases.

\(\displaystyle{\left({y}\cdot\ {y}^{{\frac{{4}}{{9}}}}\right)}^{{\frac{{1}}{{3}}}}={\left({y}^{{{1}+\frac{{4}}{{9}}}}\right)}^{{\frac{{1}}{{3}}}}\)

\(\displaystyle={\left({y}^{{\frac{{13}}{{9}}}}\right)}^{{\frac{{1}}{{3}}}}\)

\(\displaystyle={y}^{{\frac{{13}}{{27}}}}\)

The obtained expression with rational exponents can be retritten into radicals as;

\(\displaystyle{y}^{{\frac{{13}}{{27}}}}={\left({y}^{{{13}}}\right)}^{{\frac{{1}}{{27}}}}\)

\(=\sqrt[27]{y^{13}}\)

Therefore, the value of the expression is \(=\sqrt[27]{y^{13}}\)