The rationalising factor of a^{1/3} + a^{-1/3 | vedclass

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(D) Let $x = a^{1/3}$ and $y = a^{-1/3}$.We know the algebraic identity $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$.Here,$x^3 = (a^{1/3})^3 = a$ and $y^3 =

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$a^{2/3} + a^{-2/3} - 1$

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(D) Let $x = a^{1/3}$ and $y = a^{-1/3}$.We know the algebraic identity $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$.Here,$x^3 = (a^{1/3})^3 = a$ and $y^3 = (a^{-1/3})^3 = a^{-1}$.To rationalize the expression $(x + y)$,we need to multiply it by $(x^2 - xy + y^2)$ to obtain $x^3 + y^3 = a + a^{-1}$,which is a rational expression.Substituting the values of $x$ and $y$ into the factor $(x^2 - xy + y^2)$:$x^2 = (a^{1/3})^2 = a^{2/3}$$y^2 = (a^{-1/3})^2 = a^{-2/3}$$xy = a^{1/3} 	imes a^{-1/3} = a^{1/3 - 1/3} = a^0 = 1$Therefore,the rationalising factor is $a^{2/3} - 1 + a^{-2/3}$,which is $a^{2/3} + a^{-2/3} - 1$.

The rationalising factor of $a^{1/3} + a^{-1/3}$ is

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