Best solutions to - "To rationalize: Each numerator. Assume that variables represent..."

College
Calculus and Analysis
Integral Calculus
Rational exponents and radicals

sibuzwaW

Answered question

2020-12-12

To rationalize:
Each numerator. Assume that variables represent positive real numbers.
Given: An expression :

\frac{\sqrt[3]{9}}{7} .

Answer & Explanation

Caren

Skilled2020-12-13Added 96 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:
$\frac{\sqrt[3]{9}}{7}$ (Given expression)
$\Rightarrow \frac{\sqrt[3]{9}}{7} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{3}} (multiplying fraction by \sqrt[3]{3})$
$\Rightarrow \frac{\sqrt[3]{9} \cdot \sqrt[3]{3}}{7 \cdot \sqrt[3]{3}}$
$\Rightarrow \frac{\sqrt[3]{9 \cdot 3}}{7 \sqrt[3]{3}} (A p p l y i n t e g e r \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b})$
$\Rightarrow \frac{\sqrt[3]{27}}{7 \sqrt[3]{3}}$
$\Rightarrow \frac{\sqrt[3]{3^{3}}}{7 \sqrt[3]{3}}$
$\Rightarrow \frac{3}{7 \sqrt[3]{3}} (A p p l y i n t e g e r \sqrt[n]{a^{n}} = a)$
Therefore, the rationalized form of the given expression would be $\frac{3}{7 \sqrt[3]{3}} .$

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