Answer:

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Verified answer

Step-by-step explanation:

To rationalize the expression [tex]x^{1/6} + a^{1/4}[/tex], we need to eliminate the irrational exponents in the denominator. To do this, we can multiply the numerator and denominator by [tex](x^{1/6}-a^{1/4})[/tex], which is the conjugate of [tex]x^{1/6} + a^{1/4}[/tex]:

[tex]\begin{aligned} \frac{x^{1/6} + a^{1/4}}{x^{1/6}-a^{1/4}} &= \frac{(x^{1/6} + a^{1/4})(x^{1/6}-a^{1/4})}{(x^{1/6}-a^{1/4})(x^{1/6}-a^{1/4})} \\ &= \frac{x^{1/3}-x^{1/6}a^{1/4}+x^{1/6}a^{1/4}-a^{1/2}}{x^{1/3}-a^{1/2}} \end{aligned}[/tex]

Thus, [tex]\frac{x^{1/6} + a^{1/4}}{x^{1/6}-a^{1/4}}[/tex] can be simplified to [tex]\frac{x^{1/3}-x^{1/6}a^{1/4}+x^{1/6}a^{1/4}-a^{1/2}}{x^{1/3}-a^{1/2}}[/tex], which is a rational expression.