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

Falak Kinney 2021-11-06 Answered

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

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Expert Answer

Velsenw
Answered 2021-11-07 Author has 22944 answers

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

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