To rationalize: Each numerator. Assume that variables represent positive real numbers. Given: An expression : displaystylefrac{{sqrt[{{3}}]{{9}}}}{{7}}.

sibuzwaW 2020-12-12 Answered
To rationalize:
Each numerator. Assume that variables represent positive real numbers.
Given: An expression : \(\displaystyle\frac{{\sqrt[{{3}}]{{9}}}}{{7}}.\)

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Expert Answer

Caren
Answered 2020-12-13 Author has 15197 answers

Concept used:
In order to rationalize the numerator of an expression, we multiply in the numerator and denominator with such a term/expression such that we don't have any radicals left in the numerator upon simplifying.
Calcucation:
Let us rationalize the numerator as shown below:
\(\displaystyle\frac{{\sqrt[{{3}}]{{9}}}}{{7}}\) (Given expression)
\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{9}}}}{{7}}\cdot\frac{{\sqrt[{{3}}]{{3}}}}{{\sqrt[{{3}}]{{3}}}}\ \text{(multiplying fraction by}\ \displaystyle{\sqrt[{{3}}]{{3}}})\)
\(\displaystyle\Rightarrow\frac{{{\sqrt[{{3}}]{{9}}}\cdot{\sqrt[{{3}}]{{3}}}}}{{{7}\cdot{\sqrt[{{3}}]{{3}}}}}\)
\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{{9}\cdot{3}}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}{\left({A}{p}{p}{l}{y}\ integer{\sqrt[{{n}}]{{a}}}\cdot{\sqrt[{{n}}]{{b}}}={\sqrt[{{n}}]{{{a}{b}}}}\right)}\)
\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{27}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}\)
\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{{3}^{3}}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}\)
\(\displaystyle\Rightarrow\frac{3}{{{7}{\sqrt[{{3}}]{{3}}}}}{\left({A}{p}{p}{l}{y}\ integer{\sqrt[{{n}}]{{{a}^{n}}}}={a}\right)}\)
Therefore, the rationalized form of the given expression would be \(\displaystyle\frac{3}{{{7}{\sqrt[{{3}}]{{3}}}}}.\)

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