# To determine: The single radical of the expression\sqrt[3]{y\sqrt[3]{y\s

To determine: The single radical of the expression
$$\sqrt[3]{y\sqrt[3]{y\sqrt[3]{y}}}$$

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Step 1
Formula used:
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 $$=\sqrt[27]{y^{13}}$$