Discussion

Explanation:

(a + b) can be written as ${\left({\mathrm{a}}^{1/3}\right)}^3+{\left({\mathrm{b}}^{1/3}\right)}^3$

Now, ${\left({\mathrm{a}}^{1/3}\right)}^3+{\left({\mathrm{b}}^{1/3}\right)}^3$ = $\left({\mathrm{a}}^{1/3}+{\mathrm{b}}^{1/3}\right)$ × $\left({\mathrm{a}}^{2/3}-{\left(\mathrm{ab}\right)}^{1/3}+{\mathrm{b}}^{2/3}\right)$

∴ a + b =  $\left({\mathrm{a}}^{1/3}+{\mathrm{b}}^{1/3}\right)$ × $\left({\mathrm{a}}^{2/3}-{\left(\mathrm{ab}\right)}^{1/3}+{\mathrm{b}}^{2/3}\right)$

⇒ Product of $\left({\mathrm{a}}^{1/3}+{\mathrm{b}}^{1/3}\right)$ and $\left({\mathrm{a}}^{2/3}-{\left(\mathrm{ab}\right)}^{1/3}+{\mathrm{b}}^{2/3}\right)$ is rational.

∴ Rationalizing factor of $\left({\mathrm{a}}^{1/3}+{\mathrm{b}}^{1/3}\right)$ is $\left({\mathrm{a}}^{2/3}-{\left(\mathrm{ab}\right)}^{1/3}+{\mathrm{b}}^{2/3}\right)$

⇒ Rationalizing factor of $\left(2^{1/3}+3^{1/3}\right)$ is $\left(2^{2/3}-{(2\times 3)}^{1/3}+3^{2/3}\right)$ i.e., $\left(2^{2/3}-6^{1/3}+3^{2/3}\right)$

Hence, option (d). 

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