Question

the rationalising factor of a^2/3-a^1/3b^1/3+b^2/3​

Answer 1

$\textbf{Given:}$

$a^\frac{2}{3}-a^\frac{1}{3}\,b^\frac{1}{3}+b^\frac{2}{3}$

$\textbf{To find:}$

$\text{The rationalzing factor of a^\frac{2}{3}-a^\frac{1}{3}\,b^\frac{1}{3}+b^\frac{2}{3} }$

$\textbf{Solution:}$

$\text{The rationalizing factor of the given expression is an expression}$

$\text{which makes the given expression rational}$

$\text{when multiplying}$

$\text{Consider,}$

$a^\frac{2}{3}-a^\frac{1}{3}\,b^\frac{1}{3}+b^\frac{2}{3}$

$\text{when we multiply it with a^\frac{1}{3}+b^\frac{1}{3} }$

$(a^\frac{1}{3}+b^\frac{1}{3})(a^\frac{2}{3}-a^\frac{1}{3}\,b^\frac{1}{3}+b^\frac{2}{3})$

$=(a^\frac{1}{3}+b^\frac{1}{3})((a^\frac{1}{3})^2-a^\frac{1}{3}\,b^\frac{1}{3}+(b^\frac{1}{3})^2)$

$\text{Using the identity}$

$\boxed{\bf\,x^3+y^3=(x+y)(x^2-xy+y^2)}$

$\text{we get}$

$=(a^{\frac{1}{3}})^3+(b^{\frac{1}{3}})^3$

$=a+b\;\text{which is rational}$

$\therefore\textbf{The required rationalizing factor is \bf\,a^\frac{1}{3}+b^\frac{1}{3} }$