To calculate: \sqrt{\frac{3y}{20}} The single radical of the expression.

Rui Baldwin 2021-08-06 Answered
To calculate: \(\displaystyle\sqrt{{{\frac{{{3}{y}}}{{{20}}}}}}\)
The single radical of the expression.

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Expert Answer

Aamina Herring
Answered 2021-08-07 Author has 4980 answers

Formula:
\(\sqrt[n]{a}=a^{1/n}\)
\((\sqrt[n]{a})^{m}=\sqrt[n]{a^{m}}\)
\(\displaystyle={a}^{{\frac{{m}}{{n}}}}\)
The product rule for radicals is:
a) \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\)
The quotient rule for radicals is:
b) \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
Calculation:
a) \(\displaystyle\sqrt{{{\frac{{{3}{y}}}{{{20}}}}}}=\sqrt{{{\frac{{{3}{y}}}{{{5}\cdot{4}}}}}}\)
\(\displaystyle=\sqrt{{{\frac{{{3}{y}}}{{{5}\cdot{2}^{{{2}}}}}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}\sqrt{{{\frac{{{3}{y}}}{{{5}}}}}}\)
b) \(\displaystyle\sqrt{{{\frac{{{3}{y}}}{{{20}}}}}}={\frac{{{1}}}{{{2}}}}\sqrt{{{\frac{{{3}{y}}}{{{5}}}}}}\times{\frac{{\sqrt{{{5}}}}}{{\sqrt{{{5}}}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\frac{{\sqrt{{{3}}}\cdot\sqrt{{{5}}}\cdot\sqrt{{{y}}}}}{{\sqrt{{{5}}}\cdot\sqrt{{{5}}}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\frac{{\sqrt{{{15}}}\cdot\sqrt{{{y}}}}}{{{5}}}}\)
\(\displaystyle={\frac{{\sqrt{{{15}{y}}}}}{{{10}}}}\)
Hence, the simplified value of the provided expression is \(\displaystyle{\frac{{\sqrt{{{15}{y}}}}}{{{10}}}}\)

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